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Selected point groups in three dimensions

Involutional symmetry Cs, (*) =	Cyclic symmetry Cnv, (*nn) =	Dihedral symmetry Dnh, (*n22) =
Polyhedral group, , (*n32)
Tetrahedral symmetry Td, (*332) =	Octahedral symmetry Oh, (*432) =	Icosahedral symmetry Ih, (*532) =

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 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 H₃. 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 A₅ 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.

Schö.	Coxeter		Orb.	Abstract structure	Order
I			532	A5	60
Ih			*532	A5×2	120

Presentations corresponding to the above are:

${\displaystyle I:\langle s,t\mid s^{2},t^{3},(st)^{5}\rangle \ }$

${\displaystyle I_{h}:\langle s,t\mid s^{3}(st)^{-2},t^{5}(st)^{-2}\rangle .\ }$

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 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 H₃. 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 A₅ on 5 letters.

Schoe. (Orb.)	Coxeter notation	Elements	Mirror diagrams
			Orthogonal	Stereographic projection
Ih (*532)		Mirror lines: 15
I (532)		Gyration points: 125 203 302

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 edges of a spherical compound of five octahedra represent the 15 mirror planes as colored great circles. Each octahedron can represent 3 orthogonal mirror planes by its edges.
The pyritohedral symmetry is an index 5 subgroup of icosahedral symmetry, with 3 orthogonal green reflection lines and 8 red order-3 gyration points. There are 5 different orientations of pyritohedral symmetry.

The icosahedral rotation group I is of order 60. The group I is isomorphic to A₅, 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, 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 full icosahedral group I_h has order 120. It has I as normal subgroup of index 2. The group I_h is isomorphic to I × Z₂, or A₅ × Z₂, with the inversion in the center corresponding to element (identity,-1), where Z₂ is written multiplicatively.

I_h 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 D_3d and 6 versions of D_5d (symmetries like antiprisms).

I is also isomorphic to PSL₂(5), but I_h 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 P_i and Q_i act on 5 and 12 elements respectively, while the rotation matrices M_i are the elements of I. If P_k is the product of taking the permutation P_i and applying P_j to it, then for the same values of i, j and k, it is also true that Q_k is the product of taking Q_i and applying Q_j, and also that premultiplying a vector by M_k is the same as premultiplying that vector by M_i and then premultiplying that result with M_j, that is M_k = M_j × M_i. Since the permutations P_i are all the 60 even permutations of 12345, the one-to-one correspondence is made explicit, therefore the isomorphism too.

Rotation matrix	Permutation of 5 on 1 2 3 4 5	Permutation of 12 on 1 2 3 4 5 6 7 8 9 10 11 12
${\displaystyle M_{1}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}$	${\displaystyle P_{1}}$ = ()	${\displaystyle Q_{1}}$ = ()
${\displaystyle M_{2}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{2}}$ = (3 4 5)	${\displaystyle Q_{2}}$ = (1 11 8)(2 9 6)(3 5 12)(4 7 10)
${\displaystyle M_{3}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{3}}$ = (3 5 4)	${\displaystyle Q_{3}}$ = (1 8 11)(2 6 9)(3 12 5)(4 10 7)
${\displaystyle M_{4}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{4}}$ = (2 3)(4 5)	${\displaystyle Q_{4}}$ = (1 12)(2 8)(3 6)(4 9)(5 10)(7 11)
${\displaystyle M_{5}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{5}}$ = (2 3 4)	${\displaystyle Q_{5}}$ = (1 2 3)(4 5 6)(7 9 8)(10 11 12)
${\displaystyle M_{6}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{6}}$ = (2 3 5)	${\displaystyle Q_{6}}$ = (1 7 5)(2 4 11)(3 10 9)(6 8 12)
${\displaystyle M_{7}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{7}}$ = (2 4 3)	${\displaystyle Q_{7}}$ = (1 3 2)(4 6 5)(7 8 9)(10 12 11)
${\displaystyle M_{8}={\begin{bmatrix}0&-1&0\\0&0&1\\-1&0&0\end{bmatrix}}}$	${\displaystyle P_{8}}$ = (2 4 5)	${\displaystyle Q_{8}}$ = (1 10 6)(2 7 12)(3 4 8)(5 11 9)
${\displaystyle M_{9}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{9}}$ = (2 4)(3 5)	${\displaystyle Q_{9}}$ = (1 9)(2 5)(3 11)(4 12)(6 7)(8 10)
${\displaystyle M_{10}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{10}}$ = (2 5 3)	${\displaystyle Q_{10}}$ = (1 5 7)(2 11 4)(3 9 10)(6 12 8)
${\displaystyle M_{11}={\begin{bmatrix}0&0&-1\\-1&0&0\\0&1&0\end{bmatrix}}}$	${\displaystyle P_{11}}$ = (2 5 4)	${\displaystyle Q_{11}}$ = (1 6 10)(2 12 7)(3 8 4)(5 9 11)
${\displaystyle M_{12}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{12}}$ = (2 5)(3 4)	${\displaystyle Q_{12}}$ = (1 4)(2 10)(3 7)(5 8)(6 11)(9 12)
${\displaystyle M_{13}={\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix}}}$	${\displaystyle P_{13}}$ = (1 2)(4 5)	${\displaystyle Q_{13}}$ = (1 3)(2 4)(5 8)(6 7)(9 10)(11 12)
${\displaystyle M_{14}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{14}}$ = (1 2)(3 4)	${\displaystyle Q_{14}}$ = (1 5)(2 7)(3 11)(4 9)(6 10)(8 12)
${\displaystyle M_{15}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{15}}$ = (1 2)(3 5)	${\displaystyle Q_{15}}$ = (1 12)(2 10)(3 8)(4 6)(5 11)(7 9)
${\displaystyle M_{16}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{16}}$ = (1 2 3)	${\displaystyle Q_{16}}$ = (1 11 6)(2 5 9)(3 7 12)(4 10 8)
${\displaystyle M_{17}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{17}}$ = (1 2 3 4 5)	${\displaystyle Q_{17}}$ = (1 6 5 3 9)(4 12 7 8 11)
${\displaystyle M_{18}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{18}}$ = (1 2 3 5 4)	${\displaystyle Q_{18}}$ = (1 4 8 6 2)(5 7 10 12 9)
${\displaystyle M_{19}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{19}}$ = (1 2 4 5 3)	${\displaystyle Q_{19}}$ = (1 8 7 3 10)(2 12 5 6 11)
${\displaystyle M_{20}={\begin{bmatrix}0&0&1\\-1&0&0\\0&-1&0\end{bmatrix}}}$	${\displaystyle P_{20}}$ = (1 2 4)	${\displaystyle Q_{20}}$ = (1 7 4)(2 11 8)(3 5 10)(6 9 12)
${\displaystyle M_{21}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{21}}$ = (1 2 4 3 5)	${\displaystyle Q_{21}}$ = (1 2 9 11 7)(3 6 12 10 4)
${\displaystyle M_{22}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{22}}$ = (1 2 5 4 3)	${\displaystyle Q_{22}}$ = (2 3 4 7 5)(6 8 10 11 9)
${\displaystyle M_{23}={\begin{bmatrix}0&1&0\\0&0&-1\\-1&0&0\end{bmatrix}}}$	${\displaystyle P_{23}}$ = (1 2 5)	${\displaystyle Q_{23}}$ = (1 9 8)(2 6 3)(4 5 12)(7 11 10)
${\displaystyle M_{24}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{24}}$ = (1 2 5 3 4)	${\displaystyle Q_{24}}$ = (1 10 5 4 11)(2 8 9 3 12)
${\displaystyle M_{25}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{25}}$ = (1 3 2)	${\displaystyle Q_{25}}$ = (1 6 11)(2 9 5)(3 12 7)(4 8 10)
${\displaystyle M_{26}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{26}}$ = (1 3 4 5 2)	${\displaystyle Q_{26}}$ = (2 5 7 4 3)(6 9 11 10 8)
${\displaystyle M_{27}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{27}}$ = (1 3 5 4 2)	${\displaystyle Q_{27}}$ = (1 10 3 7 8)(2 11 6 5 12)
${\displaystyle M_{28}={\begin{bmatrix}-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{28}}$ = (1 3)(4 5)	${\displaystyle Q_{28}}$ = (1 7)(2 10)(3 11)(4 5)(6 12)(8 9)
${\displaystyle M_{29}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{29}}$ = (1 3 4)	${\displaystyle Q_{29}}$ = (1 9 10)(2 12 4)(3 6 8)(5 11 7)
${\displaystyle M_{30}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{30}}$ = (1 3 5)	${\displaystyle Q_{30}}$ = (1 3 4)(2 8 7)(5 6 10)(9 12 11)
${\displaystyle M_{31}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{31}}$ = (1 3)(2 4)	${\displaystyle Q_{31}}$ = (1 12)(2 6)(3 9)(4 11)(5 8)(7 10)
${\displaystyle M_{32}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{32}}$ = (1 3 2 4 5)	${\displaystyle Q_{32}}$ = (1 4 10 11 5)(2 3 8 12 9)
${\displaystyle M_{33}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{33}}$ = (1 3 5 2 4)	${\displaystyle Q_{33}}$ = (1 5 9 6 3)(4 7 11 12 8)
${\displaystyle M_{34}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{34}}$ = (1 3)(2 5)	${\displaystyle Q_{34}}$ = (1 2)(3 5)(4 9)(6 7)(8 11)(10 12)
${\displaystyle M_{35}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{35}}$ = (1 3 2 5 4)	${\displaystyle Q_{35}}$ = (1 11 2 7 9)(3 10 6 4 12)
${\displaystyle M_{36}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{36}}$ = (1 3 4 2 5)	${\displaystyle Q_{36}}$ = (1 8 2 4 6)(5 10 9 7 12)
${\displaystyle M_{37}={\begin{bmatrix}{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{37}}$ = (1 4 5 3 2)	${\displaystyle Q_{37}}$ = (1 2 6 8 4)(5 9 12 10 7)
${\displaystyle M_{38}={\begin{bmatrix}0&-1&0\\0&0&-1\\1&0&0\end{bmatrix}}}$	${\displaystyle P_{38}}$ = (1 4 2)	${\displaystyle Q_{38}}$ = (1 4 7)(2 8 11)(3 10 5)(6 12 9)
${\displaystyle M_{39}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{39}}$ = (1 4 3 5 2)	${\displaystyle Q_{39}}$ = (1 11 4 5 10)(2 12 3 9 8)
${\displaystyle M_{40}={\begin{bmatrix}-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{40}}$ = (1 4 3)	${\displaystyle Q_{40}}$ = (1 10 9)(2 4 12)(3 8 6)(5 7 11)
${\displaystyle M_{41}={\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}}}$	${\displaystyle P_{41}}$ = (1 4 5)	${\displaystyle Q_{41}}$ = (1 5 2)(3 7 9)(4 11 6)(8 10 12)
${\displaystyle M_{42}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{42}}$ = (1 4)(3 5)	${\displaystyle Q_{42}}$ = (1 6)(2 3)(4 9)(5 8)(7 12)(10 11)
${\displaystyle M_{43}={\begin{bmatrix}-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{43}}$ = (1 4 5 2 3)	${\displaystyle Q_{43}}$ = (1 9 7 2 11)(3 12 4 6 10)
${\displaystyle M_{44}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{44}}$ = (1 4)(2 3)	${\displaystyle Q_{44}}$ = (1 8)(2 10)(3 4)(5 12)(6 7)(9 11)
${\displaystyle M_{45}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{45}}$ = (1 4 2 3 5)	${\displaystyle Q_{45}}$ = (2 7 3 5 4)(6 11 8 9 10)
${\displaystyle M_{46}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{46}}$ = (1 4 2 5 3)	${\displaystyle Q_{46}}$ = (1 3 6 9 5)(4 8 12 11 7)
${\displaystyle M_{47}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{47}}$ = (1 4 3 2 5)	${\displaystyle Q_{47}}$ = (1 7 10 8 3)(2 5 11 12 6)
${\displaystyle M_{48}={\begin{bmatrix}-1&0&0\\0&1&0\\0&0&-1\end{bmatrix}}}$	${\displaystyle P_{48}}$ = (1 4)(2 5)	${\displaystyle Q_{48}}$ = (1 12)(2 9)(3 11)(4 10)(5 6)(7 8)
${\displaystyle M_{49}={\begin{bmatrix}-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{49}}$ = (1 5 4 3 2)	${\displaystyle Q_{49}}$ = (1 9 3 5 6)(4 11 8 7 12)
${\displaystyle M_{50}={\begin{bmatrix}0&0&-1\\1&0&0\\0&-1&0\end{bmatrix}}}$	${\displaystyle P_{50}}$ = (1 5 2)	${\displaystyle Q_{50}}$ = (1 8 9)(2 3 6)(4 12 5)(7 10 11)
${\displaystyle M_{51}={\begin{bmatrix}{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{51}}$ = (1 5 3 4 2)	${\displaystyle Q_{51}}$ = (1 7 11 9 2)(3 4 10 12 6)
${\displaystyle M_{52}={\begin{bmatrix}{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{52}}$ = (1 5 3)	${\displaystyle Q_{52}}$ = (1 4 3)(2 7 8)(5 10 6)(9 11 12)
${\displaystyle M_{53}={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}}$	${\displaystyle P_{53}}$ = (1 5 4)	${\displaystyle Q_{53}}$ = (1 2 5)(3 9 7)(4 6 11)(8 12 10)
${\displaystyle M_{54}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{54}}$ = (1 5)(3 4)	${\displaystyle Q_{54}}$ = (1 12)(2 11)(3 10)(4 8)(5 9)(6 7)
${\displaystyle M_{55}={\begin{bmatrix}{\frac {1}{2\phi }}&{\frac {\phi }{2}}&{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\\-{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\end{bmatrix}}}$	${\displaystyle P_{55}}$ = (1 5 4 2 3)	${\displaystyle Q_{55}}$ = (1 5 11 10 4)(2 9 12 8 3)
${\displaystyle M_{56}={\begin{bmatrix}-{\frac {\phi }{2}}&-{\frac {1}{2}}&{\frac {1}{2\phi }}\\-{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\end{bmatrix}}}$	${\displaystyle P_{56}}$ = (1 5)(2 3)	${\displaystyle Q_{56}}$ = (1 10)(2 12)(3 11)(4 7)(5 8)(6 9)
${\displaystyle M_{57}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{57}}$ = (1 5 2 3 4)	${\displaystyle Q_{57}}$ = (1 3 8 10 7)(2 6 12 11 5)
${\displaystyle M_{58}={\begin{bmatrix}{\frac {1}{2}}&{\frac {1}{2\phi }}&-{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{58}}$ = (1 5 2 4 3)	${\displaystyle Q_{58}}$ = (1 6 4 2 8)(5 12 7 9 10)
${\displaystyle M_{59}={\begin{bmatrix}{\frac {1}{2}}&-{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\{\frac {1}{2\phi }}&-{\frac {\phi }{2}}&-{\frac {1}{2}}\\{\frac {\phi }{2}}&{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}}$	${\displaystyle P_{59}}$ = (1 5 3 2 4)	${\displaystyle Q_{59}}$ = (2 4 5 3 7)(6 10 9 8 11)
${\displaystyle M_{60}={\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}}$	${\displaystyle P_{60}}$ = (1 5)(2 4)	${\displaystyle Q_{60}}$ = (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:

• S₅, the symmetric group on 5 elements
• I_h, the full icosahedral group (subject of this article, also known as H₃)
• 2I, the binary icosahedral group

They correspond to the following short exact sequences (the latter of which does not split) and product

${\displaystyle 1\to A_{5}\to S_{5}\to Z_{2}\to 1}$

${\displaystyle I_{h}=A_{5}\times Z_{2}}$

${\displaystyle 1\to Z_{2}\to 2I\to A_{5}\to 1}$

In words,

• ${\displaystyle A_{5}}$ is a normal subgroup of ${\displaystyle S_{5}}$
• ${\displaystyle A_{5}}$ is a factor of ${\displaystyle I_{h}}$, which is a direct product
• ${\displaystyle A_{5}}$ is a quotient group of ${\displaystyle 2I}$

Note that ${\displaystyle A_{5}}$ has an exceptional irreducible 3-dimensional representation (as the icosahedral rotation group), but ${\displaystyle S_{5}}$ 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:

• ${\displaystyle A_{5}\cong \operatorname {PSL} (2,5),}$ the projective special linear group, see here for a proof;
• ${\displaystyle S_{5}\cong \operatorname {PGL} (2,5),}$ the projective general linear group;
• ${\displaystyle 2I\cong \operatorname {SL} (2,5),}$ the special linear group.

Conjugacy classes

The 120 symmetries fall into 10 conjugacy classes.

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".

Schön.	Coxeter		Orb.	H-M	Structure	Cyc.	Order	Index	Mult.	Description
Ih			*532	532/m	A5×Z2		120	1	1	full group
D2h			*222	mmm	D4×D2=D2		8	15	5	fixing two opposite edges, possibly swapping them
C5v			*55	5m	D10		10	12	6	fixing a face
C3v			*33	3m	D6=S3		6	20	10	fixing a vertex
C2v			*22	2mm	D4=D2		4	30	15	fixing an edge
Cs			*	2 or m	D2		2	60	15	reflection swapping two endpoints of an edge
Th			3*2	m3	A4×Z2		24	5	5	pyritohedral group
D5d			2*5	10m2	D20=Z2×D10		20	6	6	fixing two opposite faces, possibly swapping them
D3d			2*3	3m	D12=Z2×D6		12	10	10	fixing two opposite vertices, possibly swapping them
D1d = C2h			2*	2/m	D4=Z2×D2		4	30	15	halfturn around edge midpoint, plus central inversion
S10			5×	5	Z10=Z2×Z5		10	12	6	rotations of a face, plus central inversion
S6			3×	3	Z6=Z2×Z3		6	20	10	rotations about a vertex, plus central inversion
S2			×	1	Z2		2	60	1	central inversion
I			532	532	A5		60	2	1	all rotations
T			332	332	A4		12	10	5	rotations of a contained tetrahedron
D5			522	522	D10		10	12	6	rotations around the center of a face, and h.t.s.(face)
D3			322	322	D6=S3		6	20	10	rotations around a vertex, and h.t.s.(vertex)
D2			222	222	D4=Z2		4	30	5	halfturn around edge midpoint, and h.t.s.(edge)
C5			55	5	Z5		5	24	6	rotations around a face center
C3			33	3	Z3=A3		3	40	10	rotations around a vertex
C2			22	2	Z2		2	60	15	half-turn around edge midpoint
C1			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 groups C₃
• vertex stabilizers in I_h give dihedral groups D₃
• stabilizers of an opposite pair of vertices in I give dihedral groups D₃
• stabilizers of an opposite pair of vertices in I_h give ${\displaystyle D_{3}\times \pm 1}$

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 Z₂
• edges stabilizers in I_h give Klein four-groups ${\displaystyle Z_{2}\times Z_{2}}$
• stabilizers of a pair of edges in I give Klein four-groups ${\displaystyle Z_{2}\times Z_{2}}$; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
• stabilizers of a pair of edges in I_h give ${\displaystyle Z_{2}\times Z_{2}\times Z_{2}}$; 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 antiprism they generate.

• face stabilizers in I give cyclic groups C₅
• face stabilizers in I_h give dihedral groups D₅
• stabilizers of an opposite pair of faces in I give dihedral groups D₅
• stabilizers of an opposite pair of faces in I_h give ${\displaystyle D_{5}\times \pm 1}$

Polyhedron stabilizers

For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism, ${\displaystyle I{\stackrel {\sim }{\to }}A_{5}<S_{5}}$.

• stabilizers of the inscribed tetrahedra in I are a copy of T
• stabilizers of the inscribed tetrahedra in I_h 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 I_h are a copy of T_h

Coxeter group generators

The full icosahedral symmetry group () of order 120 has generators represented by the reflection matrices R₀, R₁, R₂ below, with relations R₀ = R₁ = R₂ = (R₀×R₁) = (R₁×R₂) = (R₀×R₂) = Identity. The group () of order 60 is generated by any two of the rotations S_0,1, S_1,2, S_0,2. A rotoreflection of order 10 is generated by V_0,1,2, the product of all 3 reflections. Here ${\displaystyle \phi ={\tfrac {{\sqrt {5}}+1}{2}}}$ denotes the golden ratio.

,

	Reflections			Rotations			Rotoreflection
Name	R0	R1	R2	S0,1	S1,2	S0,2	V0,1,2
Group
Order	2	2	2	5	3	2	10
Matrix	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$	${\displaystyle \left}$
	(1,0,0)n	${\displaystyle ({\begin{smallmatrix}{\frac {\phi }{2}},{\frac {1}{2}},{\frac {\phi -1}{2}}\end{smallmatrix}})}$n	(0,1,0)n	${\displaystyle (0,-1,\phi )}$axis	${\displaystyle (1-\phi ,0,\phi )}$axis	${\displaystyle (0,0,1)}$axis

Fundamental domain

Fundamental domains for the icosahedral rotation group and the full icosahedral group are given by:

Icosahedral rotation group I

Full icosahedral group Ih

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 of the icosahedron) and the rhombic triacontahedron.

Chiral polyhedra

Class	Symbols	Picture
Archimedean	sr{5,3}
Catalan	V3.3.3.3.5

Full icosahedral symmetry

Platonic solid	Kepler–Poinsot polyhedra		Archimedean solids
{5,3}	{5/2,5}	{5/2,3}	t{5,3}	t{3,5}	r{3,5}	rr{3,5}	tr{3,5}
Platonic solid	Kepler–Poinsot polyhedra		Catalan solids
{3,5} =	{5,5/2} =	{3,5/2} =	V3.10.10	V5.6.6	V3.5.3.5	V3.4.5.4	V4.6.10

Other objects with icosahedral symmetry

• Barth surfaces
• Virus structure, and Capsid
• In chemistry, the dodecaborate ion () 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 article 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 (Klein 1888); a modern exposition is given in (Tóth 2002, Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, p. 66).

Klein's investigations continued with his discovery of order 7 and order 11 symmetries in (Klein 1878) and (Klein 1879) (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" 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.

See also

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

References

1. Sir William Rowan Hamilton (1856), "Memorandum respecting a new System of Roots of Unity" (PDF), Philosophical Magazine, 12: 446
2. Kleinert, H. & Maki, K. (1981). "Lattice Textures in Cholesteric Liquid Crystals" (PDF). Fortschritte der Physik. 29 (5): 219–259. Bibcode:1981ForPh..29..219K. doi:10.1002/prop.19810290503.

• Klein, F. (1878). "Ueber die Transformation siebenter Ordnung der elliptischen Functionen" [On the order-seven transformation of elliptic functions]. Mathematische Annalen. 14 (3): 428–471. doi:10.1007/BF01677143. S2CID 121407539. Translated in Levy, Silvio, ed. (1999). The Eightfold Way. Cambridge University Press. ISBN 978-0-521-66066-2. MR 1722410.
• Klein, F. (1879), "Ueber die Transformation elfter Ordnung der elliptischen Functionen (On the eleventh order transformation of elliptic functions)", Mathematische Annalen, 15 (3–4): 533–555, doi:10.1007/BF02086276, S2CID 120316938, collected as pp. 140–165 in Oeuvres, Tome 3
• Klein, Felix (1888), Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, Trübner & Co., ISBN 0-486-49528-0trans. George Gavin Morrice{{citation}}: CS1 maint: postscript (link)
• Tóth, Gábor (2002), Finite Möbius groups, minimal immersions of spheres, and moduli
• Peter R. Cromwell, Polyhedra (1997), p. 296
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.5 Spherical Coxeter groups

External links

• Weisstein, Eric W. "Icosahedral group". MathWorld.
• THE SUBGROUPS OF W(H3) Archived 2021-08-30 at the Wayback Machine (Subgroups of other Coxeter groups Archived 2020-08-02 at the Wayback Machine) Gotz Pfeiffer

• Finite groups
• Rotational symmetry