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In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters.


Description

Icosahedral symmetry is a mathematical property of objects indicating that an object has the same symmetries as a regular icosahedron.


As point group

Apart from the two infinite series of prismatic and antiprismatic symmetry, rotational icosahedral symmetry or chiral icosahedral symmetry of chiral objects and full icosahedral symmetry or achiral icosahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups. Icosahedral symmetry is not compatible with translational symmetry, so there are no associated crystallographic point groups or space groups. Presentations corresponding to the above are: :I: \langle s,t \mid s^2, t^3, (st)^5 \rangle\ :I_h: \langle s,t\mid s^3(st)^, t^5(st)^\rangle.\ These correspond to the icosahedral groups (rotational and full) being the (2,3,5) triangle groups. The first presentation was given by
William Rowan Hamilton Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Ire ...
in 1856, in his paper on icosian calculus. Note that other presentations are possible, for instance as an alternating group (for ''I'').


Visualizations

The full symmetry group is the Coxeter group of type . It may be represented by Coxeter notation and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group on 5 letters.


Group structure

Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120. The ''I'' is of order 60. The group ''I'' is isomorphic to ''A''5, the alternating group of even permutations of five objects. This isomorphism can be realized by ''I'' acting on various compounds, notably the compound of five cubes (which inscribe in the dodecahedron), the compound of five octahedra, or either of the two compounds of five tetrahedra (which are
enantiomorphs In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be ...
, and inscribe in the dodecahedron). The group contains 5 versions of ''T''h with 20 versions of ''D3'' (10 axes, 2 per axis), and 6 versions of ''D5''. The ''Ih'' has order 120. It has ''I'' as normal subgroup of index 2. The group ''Ih'' is isomorphic to ''I'' × ''Z''2, or ''A''5 × ''Z''2, with the inversion in the center corresponding to element (identity,-1), where ''Z''2 is written multiplicatively. ''Ih'' acts on the compound of five cubes and the compound of five octahedra, but −1 acts as the identity (as cubes and octahedra are centrally symmetric). It acts on the compound of ten tetrahedra: ''I'' acts on the two chiral halves ( compounds of five tetrahedra), and −1 interchanges the two halves. Notably, it does ''not'' act as S5, and these groups are not isomorphic; see below for details. The group contains 10 versions of ''D3d'' and 6 versions of ''D5d'' (symmetries like antiprisms). ''I'' is also isomorphic to PSL2(5), but ''Ih'' is not isomorphic to SL2(5).


Isomorphism of ''I'' with A5

It is useful to describe explicitly what the isomorphism between ''I'' and A5 looks like. In the following table, permutations Pi and Qi act on 5 and 12 elements respectively, while the rotation matrices Mi are the elements of ''I''. If Pk is the product of taking the permutation Pi and applying Pj to it, then for the same values of ''i'', ''j'' and ''k'', it is also true that Qk is the product of taking Qi and applying Qj, and also that premultiplying a vector by Mk is the same as premultiplying that vector by Mi and then premultiplying that result with Mj, that is Mk = Mj × Mi. Since the permutations Pi are all the 60 even permutations of 12345, the one-to-one correspondence is made explicit, therefore the isomorphism too. {, class="wikitable collapsible collapsed" align='center' style="font-family:'DejaVu Sans Mono','monospace'" !width="25%", Rotation matrix !width="25%", Permutation of 5
on 1 2 3 4 5 !width="50%", Permutation of 12
on 1 2 3 4 5 6 7 8 9 10 11 12 , - !M_{1}=\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix} , P_{1} = () , Q_{1} = () , - !M_{2}=\begin{bmatrix} -\frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{2} = (3 4 5) , Q_{2} = (1 11 8)(2 9 6)(3 5 12)(4 7 10) , - !M_{3}=\begin{bmatrix} -\frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{3} = (3 5 4) , Q_{3} = (1 8 11)(2 6 9)(3 12 5)(4 10 7) , - !M_{4}=\begin{bmatrix} -\frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{4} = (2 3)(4 5) , Q_{4} = (1 12)(2 8)(3 6)(4 9)(5 10)(7 11) , - !M_{5}=\begin{bmatrix} \frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{5} = (2 3 4) , Q_{5} = (1 2 3)(4 5 6)(7 9 8)(10 11 12) , - !M_{6}=\begin{bmatrix} -\frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{6} = (2 3 5) , Q_{6} = (1 7 5)(2 4 11)(3 10 9)(6 8 12) , - !M_{7}=\begin{bmatrix} \frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{7} = (2 4 3) , Q_{7} = (1 3 2)(4 6 5)(7 8 9)(10 12 11) , - !M_{8}=\begin{bmatrix} 0&-1&0\\ 0&0&1\\ -1&0&0\end{bmatrix} , P_{8} = (2 4 5) , Q_{8} = (1 10 6)(2 7 12)(3 4 8)(5 11 9) , - !M_{9}=\begin{bmatrix} -\frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{9} = (2 4)(3 5) , Q_{9} = (1 9)(2 5)(3 11)(4 12)(6 7)(8 10) , - !M_{10}=\begin{bmatrix} -\frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{10} = (2 5 3) , Q_{10} = (1 5 7)(2 11 4)(3 9 10)(6 12 8) , - !M_{11}=\begin{bmatrix} 0&0&-1\\ -1&0&0\\ 0&1&0\end{bmatrix} , P_{11} = (2 5 4) , Q_{11} = (1 6 10)(2 12 7)(3 8 4)(5 9 11) , - !M_{12}=\begin{bmatrix} \frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{12} = (2 5)(3 4) , Q_{12} = (1 4)(2 10)(3 7)(5 8)(6 11)(9 12) , - !M_{13}=\begin{bmatrix} 1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix} , P_{13} = (1 2)(4 5) , Q_{13} = (1 3)(2 4)(5 8)(6 7)(9 10)(11 12) , - !M_{14}=\begin{bmatrix} -\frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{14} = (1 2)(3 4) , Q_{14} = (1 5)(2 7)(3 11)(4 9)(6 10)(8 12) , - !M_{15}=\begin{bmatrix} -\frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{15} = (1 2)(3 5) , Q_{15} = (1 12)(2 10)(3 8)(4 6)(5 11)(7 9) , - !M_{16}=\begin{bmatrix} -\frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{16} = (1 2 3) , Q_{16} = (1 11 6)(2 5 9)(3 7 12)(4 10 8) , - !M_{17}=\begin{bmatrix} -\frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{17} = (1 2 3 4 5) , Q_{17} = (1 6 5 3 9)(4 12 7 8 11) , - !M_{18}=\begin{bmatrix} \frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{18} = (1 2 3 5 4) , Q_{18} = (1 4 8 6 2)(5 7 10 12 9) , - !M_{19}=\begin{bmatrix} -\frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{19} = (1 2 4 5 3) , Q_{19} = (1 8 7 3 10)(2 12 5 6 11) , - !M_{20}=\begin{bmatrix} 0&0&1\\ -1&0&0\\ 0&-1&0\end{bmatrix} , P_{20} = (1 2 4) , Q_{20} = (1 7 4)(2 11 8)(3 5 10)(6 9 12) , - !M_{21}=\begin{bmatrix} \frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{21} = (1 2 4 3 5) , Q_{21} = (1 2 9 11 7)(3 6 12 10 4) , - !M_{22}=\begin{bmatrix} \frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{22} = (1 2 5 4 3) , Q_{22} = (2 3 4 7 5)(6 8 10 11 9) , - !M_{23}=\begin{bmatrix} 0&1&0\\ 0&0&-1\\ -1&0&0\end{bmatrix} , P_{23} = (1 2 5) , Q_{23} = (1 9 8)(2 6 3)(4 5 12)(7 11 10) , - !M_{24}=\begin{bmatrix} -\frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{24} = (1 2 5 3 4) , Q_{24} = (1 10 5 4 11)(2 8 9 3 12) , - !M_{25}=\begin{bmatrix} -\frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{25} = (1 3 2) , Q_{25} = (1 6 11)(2 9 5)(3 12 7)(4 8 10) , - !M_{26}=\begin{bmatrix} \frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{26} = (1 3 4 5 2) , Q_{26} = (2 5 7 4 3)(6 9 11 10 8) , - !M_{27}=\begin{bmatrix} -\frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{27} = (1 3 5 4 2) , Q_{27} = (1 10 3 7 8)(2 11 6 5 12) , - !M_{28}=\begin{bmatrix} -\frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{28} = (1 3)(4 5) , Q_{28} = (1 7)(2 10)(3 11)(4 5)(6 12)(8 9) , - !M_{29}=\begin{bmatrix} -\frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{29} = (1 3 4) , Q_{29} = (1 9 10)(2 12 4)(3 6 8)(5 11 7) , - !M_{30}=\begin{bmatrix} \frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{30} = (1 3 5) , Q_{30} = (1 3 4)(2 8 7)(5 6 10)(9 12 11) , - !M_{31}=\begin{bmatrix} -\frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{31} = (1 3)(2 4) , Q_{31} = (1 12)(2 6)(3 9)(4 11)(5 8)(7 10) , - !M_{32}=\begin{bmatrix} \frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{32} = (1 3 2 4 5) , Q_{32} = (1 4 10 11 5)(2 3 8 12 9) , - !M_{33}=\begin{bmatrix} \frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{33} = (1 3 5 2 4) , Q_{33} = (1 5 9 6 3)(4 7 11 12 8) , - !M_{34}=\begin{bmatrix} \frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{34} = (1 3)(2 5) , Q_{34} = (1 2)(3 5)(4 9)(6 7)(8 11)(10 12) , - !M_{35}=\begin{bmatrix} -\frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ \frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{35} = (1 3 2 5 4) , Q_{35} = (1 11 2 7 9)(3 10 6 4 12) , - !M_{36}=\begin{bmatrix} \frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{36} = (1 3 4 2 5) , Q_{36} = (1 8 2 4 6)(5 10 9 7 12) , - !M_{37}=\begin{bmatrix} \frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{37} = (1 4 5 3 2) , Q_{37} = (1 2 6 8 4)(5 9 12 10 7) , - !M_{38}=\begin{bmatrix} 0&-1&0\\ 0&0&-1\\ 1&0&0\end{bmatrix} , P_{38} = (1 4 2) , Q_{38} = (1 4 7)(2 8 11)(3 10 5)(6 12 9) , - !M_{39}=\begin{bmatrix} -\frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{39} = (1 4 3 5 2) , Q_{39} = (1 11 4 5 10)(2 12 3 9 8) , - !M_{40}=\begin{bmatrix} -\frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{40} = (1 4 3) , Q_{40} = (1 10 9)(2 4 12)(3 8 6)(5 7 11) , - !M_{41}=\begin{bmatrix} 0&0&1\\ 1&0&0\\ 0&1&0\end{bmatrix} , P_{41} = (1 4 5) , Q_{41} = (1 5 2)(3 7 9)(4 11 6)(8 10 12) , - !M_{42}=\begin{bmatrix} \frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{42} = (1 4)(3 5) , Q_{42} = (1 6)(2 3)(4 9)(5 8)(7 12)(10 11) , - !M_{43}=\begin{bmatrix} -\frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\end{bmatrix} , P_{43} = (1 4 5 2 3) , Q_{43} = (1 9 7 2 11)(3 12 4 6 10) , - !M_{44}=\begin{bmatrix} \frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{44} = (1 4)(2 3) , Q_{44} = (1 8)(2 10)(3 4)(5 12)(6 7)(9 11) , - !M_{45}=\begin{bmatrix} \frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{45} = (1 4 2 3 5) , Q_{45} = (2 7 3 5 4)(6 11 8 9 10) , - !M_{46}=\begin{bmatrix} \frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{46} = (1 4 2 5 3) , Q_{46} = (1 3 6 9 5)(4 8 12 11 7) , - !M_{47}=\begin{bmatrix} \frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{47} = (1 4 3 2 5) , Q_{47} = (1 7 10 8 3)(2 5 11 12 6) , - !M_{48}=\begin{bmatrix} -1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix} , P_{48} = (1 4)(2 5) , Q_{48} = (1 12)(2 9)(3 11)(4 10)(5 6)(7 8) , - !M_{49}=\begin{bmatrix} -\frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\end{bmatrix} , P_{49} = (1 5 4 3 2) , Q_{49} = (1 9 3 5 6)(4 11 8 7 12) , - !M_{50}=\begin{bmatrix} 0&0&-1\\ 1&0&0\\ 0&-1&0\end{bmatrix} , P_{50} = (1 5 2) , Q_{50} = (1 8 9)(2 3 6)(4 12 5)(7 10 11) , - !M_{51}=\begin{bmatrix} \frac{1}{2\phi}&-\frac{\phi}{2}&\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{51} = (1 5 3 4 2) , Q_{51} = (1 7 11 9 2)(3 4 10 12 6) , - !M_{52}=\begin{bmatrix} \frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{52} = (1 5 3) , Q_{52} = (1 4 3)(2 7 8)(5 10 6)(9 11 12) , - !M_{53}=\begin{bmatrix} 0&1&0\\ 0&0&1\\ 1&0&0\end{bmatrix} , P_{53} = (1 5 4) , Q_{53} = (1 2 5)(3 9 7)(4 6 11)(8 12 10) , - !M_{54}=\begin{bmatrix} -\frac{\phi}{2}&-\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{54} = (1 5)(3 4) , Q_{54} = (1 12)(2 11)(3 10)(4 8)(5 9)(6 7) , - !M_{55}=\begin{bmatrix} \frac{1}{2\phi}&\frac{\phi}{2}&\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\\ -\frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\end{bmatrix} , P_{55} = (1 5 4 2 3) , Q_{55} = (1 5 11 10 4)(2 9 12 8 3) , - !M_{56}=\begin{bmatrix} -\frac{\phi}{2}&-\frac{1}{2}&\frac{1}{2\phi}\\ -\frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\end{bmatrix} , P_{56} = (1 5)(2 3) , Q_{56} = (1 10)(2 12)(3 11)(4 7)(5 8)(6 9) , - !M_{57}=\begin{bmatrix} \frac{1}{2}&-\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&\frac{1}{2\phi}\end{bmatrix} , P_{57} = (1 5 2 3 4) , Q_{57} = (1 3 8 10 7)(2 6 12 11 5) , - !M_{58}=\begin{bmatrix} \frac{1}{2}&\frac{1}{2\phi}&-\frac{\phi}{2}\\ -\frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ -\frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{58} = (1 5 2 4 3) , Q_{58} = (1 6 4 2 8)(5 12 7 9 10) , - !M_{59}=\begin{bmatrix} \frac{1}{2}&-\frac{1}{2\phi}&\frac{\phi}{2}\\ \frac{1}{2\phi}&-\frac{\phi}{2}&-\frac{1}{2}\\ \frac{\phi}{2}&\frac{1}{2}&-\frac{1}{2\phi}\end{bmatrix} , P_{59} = (1 5 3 2 4) , Q_{59} = (2 4 5 3 7)(6 10 9 8 11) , - !M_{60}=\begin{bmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\end{bmatrix} , P_{60} = (1 5)(2 4) , Q_{60} = (1 11)(2 10)(3 12)(4 9)(5 7)(6 8)


Commonly confused groups

The following groups all have order 120, but are not isomorphic: * ''S''5, the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
on 5 elements * ''Ih'', the full icosahedral group (subject of this article, also known as ''H''3) * 2''I'', the binary icosahedral group They correspond to the following short exact sequences (the latter of which does not split) and product :1\to A_5 \to S_5 \to Z_2 \to 1 :I_h = A_5 \times Z_2 :1\to Z_2 \to 2I\to A_5 \to 1 In words, * A_5 is a '' normal subgroup'' of S_5 * A_5 is a ''factor'' of I_h, which is a '' direct product'' * A_5 is a ''
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
'' of 2I Note that A_5 has an exceptional irreducible 3-dimensional
representation Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
(as the icosahedral rotation group), but S_5 does not have an irreducible 3-dimensional representation, corresponding to the full icosahedral group not being the symmetric group. These can also be related to linear groups over the finite field with five elements, which exhibit the subgroups and covering groups directly; none of these are the full icosahedral group: * A_5 \cong \operatorname{PSL}(2,5), the projective special linear group, see here for a proof; * S_5 \cong \operatorname{PGL}(2,5), the projective general linear group; * 2I \cong \operatorname{SL}(2,5), the special linear group.


Conjugacy classes

The 120 symmetries fall into 10 conjugacy classes. {, class=wikitable , + conjugacy classes !''I'' !additional classes of ''Ih'' , - , * identity, order 1 * 12 × rotation by ±72°, order 5, around the 6 axes through the face centers of the dodecahedron * 12 × rotation by ±144°, order 5, around the 6 axes through the face centers of the dodecahedron * 20 × rotation by ±120°, order 3, around the 10 axes through vertices of the dodecahedron * 15 × rotation by 180°, order 2, around the 15 axes through midpoints of edges of the dodecahedron , * central inversion, order 2 * 12 × rotoreflection by ±36°, order 10, around the 6 axes through the face centers of the dodecahedron * 12 × rotoreflection by ±108°, order 10, around the 6 axes through the face centers of the dodecahedron * 20 × rotoreflection by ±60°, order 6, around the 10 axes through the vertices of the dodecahedron * 15 × reflection, order 2, at 15 planes through edges of the dodecahedron


Subgroups of the full icosahedral symmetry group

Each line in the following table represents one class of conjugate (i.e., geometrically equivalent) subgroups. The column "Mult." (multiplicity) gives the number of different subgroups in the conjugacy class. Explanation of colors: green = the groups that are generated by reflections, red = the chiral (orientation-preserving) groups, which contain only rotations. The groups are described geometrically in terms of the dodecahedron. The abbreviation "h.t.s.(edge)" means "halfturn swapping this edge with its opposite edge", and similarly for "face" and "vertex". {, class="wikitable sortable" ! Schön., , colspan=2, Coxeter, , Orb., , H-M, , Structure, , Cyc., ,
Order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
, , Index, , Mult., , Description , - align=center BGCOLOR="#e0f0f0" , Ih, , ,3, , , *532, , 2/m, , A5×Z2, , , , 120, , 1, , 1, , full group , - align=center BGCOLOR="#e0f0f0" , D2h, , ,2, , , *222, , mmm, , D4×D2=D23, , , , 8, , 15, , 5, , fixing two opposite edges, possibly swapping them , -align=center BGCOLOR="#e0f0f0" , C5v , , , , , *55 , , 5m, , D10, , , , 10 , , 12, , 6, , fixing a face , -align=center BGCOLOR="#e0f0f0" , C3v , , , , , *33 , , 3m, , D6=S3, , , , 6 , , 20, , 10, , fixing a vertex , -align=center BGCOLOR="#e0f0f0" , C2v , , , , , *22 , , 2mm, , D4=D22, , , , 4 , , 30, , 15, , fixing an edge , -align=center BGCOLOR="#e0f0f0" , Cs , , nbsp;, , , * , , or m, , D2, , , , 2 , , 60, , 15, , reflection swapping two endpoints of an edge , - align=center BGCOLOR="#f0f0e0" , Th, , +,4, , , 3*2, , m, , A4×Z2, , , , 24, , 5, , 5, , pyritohedral group , -align=center BGCOLOR="#f0f0e0" , D5d , , +,10, , , 2*5 , , m2, , D20=Z2×D10, , , , 20 , , 6, , 6, , fixing two opposite faces, possibly swapping them , -align=center BGCOLOR="#f0f0e0" , D3d , , +,6, , , 2*3 , , m, , D12=Z2×D6, , , , 12 , , 10, , 10, , fixing two opposite vertices, possibly swapping them , -align=center BGCOLOR="#f0f0e0" , D1d = C2h , , +,2, , , 2* , , 2/m, , D4= Z2×D2, , , , 4 , , 30, , 15, , halfturn around edge midpoint, plus central inversion , -align=center BGCOLOR="#e0e0e0" , S10 , , +,10+, , , 5× , , , , Z10=Z2×Z5, , , , 10 , , 12, , 6, , rotations of a face, plus central inversion , -align=center BGCOLOR="#e0e0e0" , S6 , , +,6+, , , 3× , , , , Z6=Z2×Z3, , , , 6 , , 20, , 10, , rotations about a vertex, plus central inversion , -align=center BGCOLOR="#e0e0e0" , S2 , , +,2+, , , × , , , , Z2, , , , 2 , , 60, , 1, , central inversion , -align=center BGCOLOR="#f0e0f0" , I, , ,3sup>+, , , , 532, , 532, , A5, , , , 60, , 2, , 1, , all rotations , - align=center BGCOLOR="#f0e0f0" , T, , ,3sup>+, , , , 332, , 332, , A4 , , , , 12, , 10, , 5, , rotations of a contained tetrahedron , - align=center BGCOLOR="#f0e0f0" , D5, , ,5sup>+, , , , 522, , 522, , D10, , , , 10, , 12, , 6, , rotations around the center of a face, and h.t.s.(face) , - align=center BGCOLOR="#f0e0f0" , D3, , ,3sup>+, , , , 322, , 322, , D6=S3, , , , 6, , 20, , 10, , rotations around a vertex, and h.t.s.(vertex) , - align=center BGCOLOR="#f0e0f0" , D2, , ,2sup>+, , , , 222, , 222, , D4=Z22, , , , 4, , 30, , 15, , halfturn around edge midpoint, and h.t.s.(edge) , - align=center BGCOLOR="#f0e0f0" , C5, , sup>+, , , , 55, , 5, , Z5, , , , 5, , 24, , 6, , rotations around a face center , - align=center BGCOLOR="#f0e0f0" , C3, , sup>+, , , , 33, , 3, , Z3=A3, , , , 3, , 40, , 10, , rotations around a vertex , - align=center BGCOLOR="#f0e0f0" , C2, , sup>+, , , , 22, , 2, , Z2, , , , 2, , 60, , 15, , half-turn around edge midpoint , - align=center BGCOLOR="#f0e0f0" , C1, , nbsp;sup>+, , , , 11, , 1, , Z1, , , , 1, , 120, , 1, , trivial group


Vertex stabilizers

Stabilizers of an opposite pair of vertices can be interpreted as stabilizers of the axis they generate. * vertex stabilizers in ''I'' give
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
s ''C''3 * vertex stabilizers in ''Ih'' give dihedral groups ''D''3 * stabilizers of an opposite pair of vertices in ''I'' give dihedral groups ''D''3 * stabilizers of an opposite pair of vertices in ''Ih'' give D_3 \times \pm 1


Edge stabilizers

Stabilizers of an opposite pair of edges can be interpreted as stabilizers of the rectangle they generate. * edges stabilizers in ''I'' give cyclic groups ''Z''2 * edges stabilizers in ''Ih'' give Klein four-groups Z_2 \times Z_2 * stabilizers of a pair of edges in ''I'' give Klein four-groups Z_2 \times Z_2; there are 5 of these, given by rotation by 180° in 3 perpendicular axes. * stabilizers of a pair of edges in ''Ih'' give Z_2 \times Z_2 \times Z_2; there are 5 of these, given by reflections in 3 perpendicular axes.


Face stabilizers

Stabilizers of an opposite pair of faces can be interpreted as stabilizers of the
anti-prism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...
they generate. * face stabilizers in ''I'' give cyclic groups ''C''5 * face stabilizers in ''Ih'' give dihedral groups ''D''5 * stabilizers of an opposite pair of faces in ''I'' give dihedral groups ''D''5 * stabilizers of an opposite pair of faces in ''Ih'' give D_5 \times \pm 1


Polyhedron stabilizers

For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, I \stackrel{\sim}\to A_5 < S_5. * stabilizers of the inscribed tetrahedra in ''I'' are a copy of ''T'' * stabilizers of the inscribed tetrahedra in ''Ih'' are a copy of ''T'' * stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedra) in ''I'' are a copy of ''T'' * stabilizers of the inscribed cubes (or opposite pair of tetrahedra, or octahedra) in ''Ih'' are a copy of ''Th''


Coxeter group generators

The full icosahedral symmetry group ,3() of order 120 has generators represented by the reflection matrices R0, R1, R2 below, with relations R02 = R12 = R22 = (R0×R1)5 = (R1×R2)3 = (R0×R2)2 = Identity. The group ,3sup>+ () of order 60 is generated by any two of the rotations S0,1, S1,2, S0,2. A rotoreflection of order 10 is generated by V0,1,2, the product of all 3 reflections. Here \phi = \tfrac {\sqrt{5}+1} {2} denotes the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...
. {, class=wikitable , + ,3 ! !colspan=3, Reflections !colspan=3, Rotations !Rotoreflection , - !Name ! R0 ! R1 ! R2 ! S0,1 ! S1,2 ! S0,2 ! V0,1,2 , - align=center !Group , , , , , , , , - align=center !Order , 2, , 2, , 2, , 5, , 3, , 2, , 10 , - align=center !Matrix , \left \begin{smallmatrix} -1&0&0\\ 0&1&0\\ 0&0&1\end{smallmatrix} \right/math> , \left[ \begin{smallmatrix} {\frac {1-\phi}{2&{\frac {-\phi}{2&{\frac {-1}{2\\ {\frac {-\phi}{2&{\frac {1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {1-\phi}{2&{\frac {\phi}{2\end{smallmatrix} \right] , \left[ \begin{smallmatrix} 1&0&0\\ 0&-1&0\\ 0&0&1\end{smallmatrix} \right] , \left[ \begin{smallmatrix} {\frac {\phi-1}{2&{\frac {\phi}{2&{\frac {1}{2\\ {\frac {-\phi}{2&{\frac {1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {1-\phi}{2&{\frac {\phi}{2\end{smallmatrix} \right] , \left[ \begin{smallmatrix} {\frac {1-\phi}{2&{\frac {\phi}{2&{\frac {-1}{2\\ {\frac {-\phi}{2&{\frac {-1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {\phi-1}{2&{\frac {\phi}{2\end{smallmatrix} \right] , \left[ \begin{smallmatrix} -1&0&0\\ 0&-1&0\\ 0&0&1\end{smallmatrix} \right] , \left[ \begin{smallmatrix} {\frac {\phi-1}{2&{\frac {-\phi}{2&{\frac {1}{2\\ {\frac {-\phi}{2&{\frac {-1}{2&{\frac {1-\phi}{2\\ {\frac {-1}{2&{\frac {\phi-1}{2&{\frac {\phi}{2\end{smallmatrix} \right] , - align=center ! , (1,0,0)n , ( \begin{smallmatrix}\frac {\phi}{2}, \frac {1}{2}, \frac {\phi-1}{2}\end{smallmatrix} )n , (0,1,0)n , (0,-1,\phi)axis , (1-\phi,0,\phi)axis , (0,0,1)axis ,


Fundamental domain

Fundamental domains for the icosahedral rotation group and the full icosahedral group are given by: {, class=wikitable width=580 , - align=center valign=top ,
Icosahedral rotation group
''I'' ,
Full icosahedral group
''I''h ,
Faces of disdyakis triacontahedron are the fundamental domain In the disdyakis triacontahedron one full face is a fundamental domain; other solids with the same symmetry can be obtained by adjusting the orientation of the faces, e.g. flattening selected subsets of faces to combine each subset into one face, or replacing each face by multiple faces, or a curved surface.


Polyhedra with icosahedral symmetry

Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
of the icosahedron) and the rhombic triacontahedron.


Chiral polyhedra

{, class=wikitable !Class ! Symbols ! Picture , -align=center ! Archimedean ! sr{5,3}
, 50px , -align=center !
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! V3.3.3.3.5
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Full icosahedral symmetry

{, class=wikitable !colspan=1, Platonic solid, , colspan=2, Kepler–Poinsot polyhedra !colspan=5, Archimedean solids , - align=center ,
{5,3}
,
{5/2,5}
,
{5/2,3}
,
t{5,3}
,
t{3,5}
,
r{3,5}
,
rr{3,5}
,
tr{3,5}
, - align=center !Platonic solid, , colspan=2, Kepler–Poinsot polyhedra !colspan=5, Catalan solids , - align=center ,
{3,5}
= ,
{5,5/2}
= ,
{3,5/2}
= ,
V3.10.10
,
V5.6.6
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V3.5.3.5
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V3.4.5.4
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V4.6.10


Other objects with icosahedral symmetry

* Barth surfaces * Virus structure, and
Capsid A capsid is the protein shell of a virus, enclosing its genetic material. It consists of several oligomeric (repeating) structural subunits made of protein called protomers. The observable 3-dimensional morphological subunits, which may or may ...
* In chemistry, the dodecaborate ion ( 12H12sup>2−) and the dodecahedrane molecule (C20H20)


Liquid crystals with icosahedral symmetry

For the intermediate material phase called liquid crystals the existence of icosahedral symmetry was proposed by H. Kleinert and K. Maki and its structure was first analyzed in detail in that paper. See the review articl
here
In aluminum, the icosahedral structure was discovered experimentally three years after this by Dan Shechtman, which earned him the Nobel Prize in 2011.


Related geometries

Icosahedral symmetry is equivalently the projective special linear group PSL(2,5), and is the symmetry group of the modular curve X(5), and more generally PSL(2,''p'') is the symmetry group of the modular curve X(''p''). The modular curve X(5) is geometrically a dodecahedron with a cusp at the center of each polygonal face, which demonstrates the symmetry group. This geometry, and associated symmetry group, was studied by Felix Klein as the monodromy groups of a Belyi surface – a Riemann surface with a holomorphic map to the Riemann sphere, ramified only at 0, 1, and infinity (a Belyi function) – the cusps are the points lying over infinity, while the vertices and the centers of each edge lie over 0 and 1; the degree of the covering (number of sheets) equals 5. This arose from his efforts to give a geometric setting for why icosahedral symmetry arose in the solution of the quintic equation, with the theory given in the famous ; a modern exposition is given in . Klein's investigations continued with his discovery of order 7 and order 11 symmetries in and (and associated coverings of degree 7 and 11) and dessins d'enfants, the first yielding the Klein quartic, whose associated geometry has a tiling by 24 heptagons (with a cusp at the center of each). Similar geometries occur for PSL(2,''n'') and more general groups for other modular curves. More exotically, there are special connections between the groups PSL(2,5) (order 60), PSL(2,7) (order 168) and PSL(2,11) (order 660), which also admit geometric interpretations – PSL(2,5) is the symmetries of the icosahedron (genus 0), PSL(2,7) of the Klein quartic (genus 3), and PSL(2,11) the buckyball surface (genus 70). These groups form a "
trinity The Christian doctrine of the Trinity (, from 'threefold') is the central dogma concerning the nature of God in most Christian churches, which defines one God existing in three coequal, coeternal, consubstantial divine persons: God the ...
" in the sense of Vladimir Arnold, which gives a framework for the various relationships; see '' trinities'' for details. There is a close relationship to other
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
.


See also

* Tetrahedral symmetry * Octahedral symmetry * Binary icosahedral group * Icosian calculus


References

* Translated in * * * * Peter R. Cromwell, ''Polyhedra'' (1997), p. 296 * ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, * ''Kaleidoscopes: Selected Writings of
H.S.M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
'', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,

* Norman Johnson (mathematician), N.W. Johnson: ''Geometries and Transformations'', (2018) Chapter 11: ''Finite symmetry groups'', 11.5 Spherical Coxeter groups


External links

*
THE SUBGROUPS OF W(H3)


{{Webarchive, url=https://web.archive.org/web/20200802022826/http://schmidt.nuigalway.ie/subgroups/cox.html , date=2020-08-02 ) Gotz Pfeiffer Finite groups Rotational symmetry