A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is the dual of the icosahedron.

The group of symmetries that includes reflections is S₅, or the group of permutations of five objects, since there is exactly one such symmetry for each permutation of the five pairs of diametrically opposite vertices. The set of orientation-preserving symmetries forms a group referred to as A₅.

Details

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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. The icosahedral rotation group I is of order 60. The group I is isomorphic to A₅, the alternating group on 5 objects. 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 × C₂, or A₅ × C₂, with the inversion in the center corresponding to element (identity,-1), where C₂ is written multiplicatively. The group contains 10 versions of D_3d and 6 versions of D_5d (symmetries like antiprisms).

Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order
I	*+	532	60
Ih	*	*532	120

Presentations

I: $\backslash langle\; s,t\; \backslash mid\; s^2,\; t^3,\; (st)^5\; \backslash rangle$

I_h: $\backslash langle\; s,t\backslash mid\; s^3(st)^\{-2\},\; t^5(st)^\{-2\}\backslash rangle$

Note that other presentations are possible.

Conjugacy classes

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The conjugacy classes of I are:

• identity
• 12 × rotation by 72°
• 12 × rotation by 144°
• 20 × rotation by 120°
• 15 × rotation by 180°

Those of I_h include also each with inversion:

• inversion
• 12 × rotoreflection by 108°
• 12 × rotoreflection by 36°
• 20 × rotoreflection by 60°
• 15 × reflection

Subgroups

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I contains 5 copies of T.

I_h contains 5 copies of T_h.

Solids with full icosahedral symmetry

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(For details see below.)

Platonic solids - regular polyhedra (all faces of the same type)

{5,3}

{3,5}

Archimedean solids - polyhedra with more than one polygon face type.

3.10.10
4.6.10
5.6.6
3.4.5.4
3.5.3.5

Catalan solids - duals of the Archimedean solids.

V3.10.10
V4.6.10
V5.6.6
V3.4.5.4
V3.5.3.5

Platonic solids

(dodecahedron.gif)
(icosahedron.gif)

Name	Picture	Faces	Edges	Vertices	Edges per face	Faces meeting at each vertex
dodecahedron	12	30	20	5	3
icosahedron	20	30	12	3	5

Achiral Archimedean solids

(icosidodecahedron.gif)
(truncateddodecahedron.gif)
(truncatedicosahedron.gif)
(rhombicosidodecahedron.gif)
(truncatedicosidodecahedron.gif)

Name	picture	Faces		Edges	Vertices	Vertex configuration
icosidodecahedron (quasi-regular: vertex- and edge-uniform)	32	20 triangles 12 pentagons	60	30	3,5,3,5
truncated dodecahedron	32	20 triangles 12 decagons	90	60	3,10,10
truncated icosahedron or commonly football (soccer ball)	32	12 pentagons 20 hexagons	90	60	5,6,6
rhombicosidodecahedron or small rhombicosidodecahedron	62	20 triangles 30 squares 12 pentagons	120	60	3,4,5,4
truncated icosidodecahedron or great rhombicosidodecahedron	62	30 squares 20 hexagons 12 decagons	180	120	4,6,10

Achiral Catalan solids

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon
rhombic triacontahedron (quasi-regular dual: face- and edge-uniform) (rhombictriacontahedron.gif)	icosidodecahedron	30	60	32	rhombus
triakis icosahedron (triakisicosahedron.gif)	truncated dodecahedron	60	90	32	isosceles triangle
pentakis dodecahedron (pentakisdodecahedron.gif)	truncated icosahedron	60	90	32	isosceles triangle
deltoidal hexecontahedron (deltoidalhexecontahedronahedron.gif)	rhombicosidodecahedron	60	120	62	kite
disdyakis triacontahedron or hexakis icosahedron (disdyakistriacontahedron.gif)	truncated icosidodecahedron	120	180	62	scalene triangle

Kepler-Poinsot solids

Finite groups | Symmetry

Achiral nonconvex uniform polyhedra

Chiral Archimedean and Catalan solids

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Archimedean solids:

(snubdodecahedronccw.gif)
snubdodecahedroncw.jpg
(snubdodecahedroncw.gif)

Name	picture	Faces		Edges	Vertices	Vertex configuration
snub dodecahedron or snub icosidodecahedron (2 chiral forms)	92	80 triangles 12 pentagons	150	60	3,3,3,3,5

Catalan solids:

Name	picture	Dual Archimedean solid	Faces	Edges	Vertices	Face Polygon
pentagonal hexecontahedron | pentagonalhexecontahedroncw.jpg (pentagonalhexecontahedronccw.gif)(pentagonalhexecontahedroncw.gif)	snub dodecahedron	60	150	92	irregular pentagon

Stellated Archimedean solids:

• the snub dodecadodecahedron
• the great inverted snub icosidodecahedron or great vertisnub icosidodecahedron
• the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron

Chiral nonconvex uniform polyhedra

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See also

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• Polyhedral compound