# Icosahedral symmetry

Selected point groups in three dimensions :
| Involutional symmetry C$_s$, (*) [ ] = | Cyclic symmetry C$_{nv}$, (*nn) [n] = | Dihedral symmetry D$_{nh}$, (*n22) [n,2] = |
| Polyhedral group, [n,3], (*n32) |||
| Tetrahedral symmetry T$_d$, (*332) [3,3] = | Octahedral symmetry O$_h$, (*432) [4,3] = | Icosahedral symmetry I$_h$, (*532) [5,3] = |
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 H$_3$. It may be represented by Coxeter notation [5,3] and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A$_5$ on 5 letters.

## 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 | Coxeter | Orb. | Abstract structure | Order |
| I | [5,3]$^+$ | | 532 | A$_5$ | 60 |
| I$_h$ | [5,3] | | *532 | A$_5$×2 | 120 |
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)^{-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 H$_3$. It may be represented by Coxeter notation [5,3] and Coxeter diagram . The set of rotational symmetries forms a subgroup that is isomorphic to the alternating group A$_5$ on 5 letters.
| Schoe. (Orb.) | Coxeter notation | Elements | Mirror diagrams | Mirror diagrams | Mirror diagrams | Mirror diagrams |
| Schoe. (Orb.) | Coxeter notation | Elements | Orthogonal | Stereographic projection | Stereographic projection | Stereographic projection |
| I$_h$ (*532) | [5,3] | Mirror lines: 15 | | | | |
| I (532) | [5,3]$^+$ | Gyration points: 12$_5$ 20$_3$ 30$_2$ | | | | |

## 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 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, and inscribe in the dodecahedron). The group contains 5 versions of T$_h$ with 20 versions of D$_3$ (10 axes, 2 per axis), and 6 versions of D$_5$.
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 × Z$_2$, or A$_5$ × Z$_2$, with the inversion in the center corresponding to element (identity,-1), where Z$_2$ 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 S$_5$, 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 PSL$_2$(5), but I$_h$ is not isomorphic to SL$_2$(5).

### Isomorphism of I with A$_5$

It is useful to describe explicitly what the isomorphism between I and A$_5$ 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 |
| $M_{1}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}$ | $P_{1}$ = | $Q_{1}$ = |
| $M_{2}={\begin{bmatrix}-{\frac {1}{2}}&{\frac {1}{2\phi }}&{\frac {\phi }{2}}\\-{\frac {1}{2\phi }}&{\frac {\phi }{2}}&-{\frac {1}{2}}\\-{\frac {\phi }{2}}&-{\frac {1}{2}}&-{\frac {1}{2\phi }}\end{bmatrix}}$ | $P_{2}$ = (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 on 5 elements
* I$_h$, the full icosahedral group (subject of this article, also known as H$_3$)
* 2I, 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 of $2I$
Note that $A_{5}$ has an exceptional irreducible 3-dimensional representation (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.
conjugacy classes :
| I | additional classes of I$_h$ |
| 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".
| Schön. | Coxeter | Orb. | H-M | Structure | Order | Index | Mult. | Description |
| I$_h$ | [5,3] | *532 | 532/m | A$_5$×Z$_2$ | 120 | 1 | 1 | full group |
| D$_{2h}$ | [2,2] | *222 | mmm | D$_4$×D$_2$=D$_2$$^3$ | 8 | 15 | 5 | fixing two opposite edges, possibly swapping them |
| C$_{5v}$ | [5] | *55 | 5m | D$_{10}$ | 10 | 12 | 6 | fixing a face |
| C$_{3v}$ | [3] | *33 | 3m | D$_6$=S$_3$ | 6 | 20 | 10 | fixing a vertex |
| C$_{2v}$ | [2] | *22 | 2mm | D$_4$=D$_2$$^2$ | 4 | 30 | 15 | fixing an edge |
| C$_s$ | [ ] | * | 2 or m | D$_2$ | 2 | 60 | 15 | reflection swapping two endpoints of an edge |
| T$_h$ | [3$^+$,4] | 3*2 | m3 | A$_4$×Z$_2$ | 24 | 5 | 5 | pyritohedral group |
| D$_{5d}$ | [2$^+$,10] | 2*5 | 10m2 | D$_{20}$=Z$_2$×D$_{10}$ | 20 | 6 | 6 | fixing two opposite faces, possibly swapping them |
| D$_{3d}$ | [2$^+$,6] | 2*3 | 3m | D$_{12}$=Z$_2$×D$_6$ | 12 | 10 | 10 | fixing two opposite vertices, possibly swapping them |
| D$_{1d}$ = C$_{2h}$ | [2$^+$,2] | 2* | 2/m | D$_4$=Z$_2$×D$_2$ | 4 | 30 | 15 | halfturn around edge midpoint, plus central inversion |
| S$_{10}$ | [2$^+$,10$^+$] | 5× | 5 | Z$_{10}$=Z$_2$×Z$_5$ | 10 | 12 | 6 | rotations of a face, plus central inversion |
| S$_6$ | [2$^+$,6$^+$] | 3× | 3 | Z$_6$=Z$_2$×Z$_3$ | 6 | 20 | 10 | rotations about a vertex, plus central inversion |
| S$_2$ | [2$^+$,2$^+$] | × | 1 | Z$_2$ | 2 | 60 | 1 | central inversion |
| I | [5,3]$^+$ | 532 | 532 | A$_5$ | 60 | 2 | 1 | all rotations |
| T | [3,3]$^+$ | 332 | 332 | A$_4$ | 12 | 10 | 5 | rotations of a contained tetrahedron |
| D$_5$ | [2,5]$^+$ | 522 | 522 | D$_{10}$ | 10 | 12 | 6 | rotations around the center of a face, and h.t.s.(face) |
| D$_3$ | [2,3]$^+$ | 322 | 322 | D$_6$=S$_3$ | 6 | 20 | 10 | rotations around a vertex, and h.t.s.(vertex) |
| D$_2$ | [2,2]$^+$ | 222 | 222 | D$_4$=Z$_2$$^2$ | 4 | 30 | 5 | halfturn around edge midpoint, and h.t.s.(edge) |
| C$_5$ | [5]$^+$ | 55 | 5 | Z$_5$ | 5 | 24 | 6 | rotations around a face center |
| C$_3$ | [3]$^+$ | 33 | 3 | Z$_3$=A$_3$ | 3 | 40 | 10 | rotations around a vertex |
| C$_2$ | [2]$^+$ | 22 | 2 | Z$_2$ | 2 | 60 | 15 | half-turn around edge midpoint |
| C$_1$ | [ ]$^+$ | 11 | 1 | Z$_1$ | 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 C$_3$
* vertex stabilizers in I$_h$ 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 I$_h$ 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 I$_h$ 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 I$_h$ 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 antiprism they generate.
* face stabilizers in I give cyclic groups C$_5$
* face stabilizers in I$_h$ 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 I$_h$ 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 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 [5,3] of order 120 has generators represented by the reflection matrices R$_0$, R$_1$, R$_2$ below, with relations R$_0$$^2$ = R$_1$$^2$ = R$_2$$^2$ = (R$_0$×R$_1$)$^5$ = (R$_1$×R$_2$)$^3$ = (R$_0$×R$_2$)$^2$ = Identity. The group [5,3]$^+$ of order 60 is generated by any two of the rotations S$_{0,1}$, S$_{1,2}$, S$_{0,2}$. A rotoreflection of order 10 is generated by V$_{0,1,2}$, the product of all 3 reflections. Here $\phi ={\tfrac {{\sqrt {5}}+1}{2}}$ denotes the golden ratio.
[5,3], :
| | Reflections | Reflections | Reflections | Rotations | Rotations | Rotations | Rotoreflection |
| Name | R$_0$ | R$_1$ | R$_2$ | S$_{0,1}$ | S$_{1,2}$ | S$_{0,2}$ | V$_{0,1,2}$ |
| Group | | | | | | | |
| Order | 2 | 2 | 2 | 5 | 3 | 2 | 10 |
| Matrix | $\left[{\begin{smallmatrix}-1&0&0\\0&1&0\\0&0&1\end{smallmatrix}}\right]$ | $\left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {-\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$ | $\left[{\begin{smallmatrix}1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]$ | $\left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {1-\phi }{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$ | $\left[{\begin{smallmatrix}{\frac {1-\phi }{2}}&{\frac {\phi }{2}}&{\frac {-1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$ | $\left[{\begin{smallmatrix}-1&0&0\\0&-1&0\\0&0&1\end{smallmatrix}}\right]$ | $\left[{\begin{smallmatrix}{\frac {\phi -1}{2}}&{\frac {-\phi }{2}}&{\frac {1}{2}}\\{\frac {-\phi }{2}}&{\frac {-1}{2}}&{\frac {1-\phi }{2}}\\{\frac {-1}{2}}&{\frac {\phi -1}{2}}&{\frac {\phi }{2}}\end{smallmatrix}}\right]$ |
| | (1,0,0)$_n$ | $({\begin{smallmatrix}{\frac {\phi }{2}},{\frac {1}{2}},{\frac {\phi -1}{2}}\end{smallmatrix}})$$_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:
| 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 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 | Kepler–Poinsot polyhedra | Archimedean solids | Archimedean solids | Archimedean solids | Archimedean solids | 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 | Kepler–Poinsot polyhedra | Catalan solids | Catalan solids | Catalan solids | Catalan solids | 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 ([B$_{12}$H$_{12}$]$^{2−}$) and the dodecahedrane molecule (C$_{20}$H$_{20}$)

### 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.