
In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same
symmetries as a
regular icosahedron
The regular icosahedron (or simply ''icosahedron'') is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with Regular polygon, regular faces to each of its pentagonal faces, or by putting ...
. Examples of other
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
with icosahedral symmetry include the
regular dodecahedron (the
dual of the icosahedron) and the
rhombic triacontahedron
The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombus, rhombic face (geometry), faces. It has 60 edge (geometry), edges and 32 vertex ...
.
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
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
is the
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
of type . It may be represented by
Coxeter notation and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
. The set of rotational symmetries forms a subgroup that is isomorphic to the
alternating group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
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 group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
s.
Icosahedral symmetry is not compatible with
translational symmetry, so there are no associated
crystallographic point groups or
space group
In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that ...
s.
Presentations corresponding to the above are:
:
:
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 (4 August 1805 – 2 September 1865) was an Irish astronomer, mathematician, and physicist who made numerous major contributions to abstract algebra, classical mechanics, and optics. His theoretical works and mathema ...
in 1856, in his paper on
icosian calculus.
Note that other presentations are possible, for instance as an
alternating group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
(for ''I'').
Visualizations
The full
symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...
is the
Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...
of type . It may be represented by
Coxeter notation and
Coxeter diagram
Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century.
Coxeter was born in England and educated ...
. The set of rotational symmetries forms a subgroup that is isomorphic to the
alternating group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
on 5 letters.
Group structure
Every
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
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
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to ''A''
5, the
alternating group
In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...
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
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
), 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 ''I
h'' has order 120. It has ''I'' as
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
of
index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
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 A5
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
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...
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
, -
!
,
= ()
,
= ()
, -
!
,
= (3 4 5)
,
= (1 11 8)(2 9 6)(3 5 12)(4 7 10)
, -
!
,
= (3 5 4)
,
= (1 8 11)(2 6 9)(3 12 5)(4 10 7)
, -
!
,
= (2 3)(4 5)
,
= (1 12)(2 8)(3 6)(4 9)(5 10)(7 11)
, -
!
,
= (2 3 4)
,
= (1 2 3)(4 5 6)(7 9 8)(10 11 12)
, -
!
,
= (2 3 5)
,
= (1 7 5)(2 4 11)(3 10 9)(6 8 12)
, -
!
,
= (2 4 3)
,
= (1 3 2)(4 6 5)(7 8 9)(10 12 11)
, -
!
,
= (2 4 5)
,
= (1 10 6)(2 7 12)(3 4 8)(5 11 9)
, -
!
,
= (2 4)(3 5)
,
= (1 9)(2 5)(3 11)(4 12)(6 7)(8 10)
, -
!
,
= (2 5 3)
,
= (1 5 7)(2 11 4)(3 9 10)(6 12 8)
, -
!
,
= (2 5 4)
,
= (1 6 10)(2 12 7)(3 8 4)(5 9 11)
, -
!
,
= (2 5)(3 4)
,
= (1 4)(2 10)(3 7)(5 8)(6 11)(9 12)
, -
!
,
= (1 2)(4 5)
,
= (1 3)(2 4)(5 8)(6 7)(9 10)(11 12)
, -
!
,
= (1 2)(3 4)
,
= (1 5)(2 7)(3 11)(4 9)(6 10)(8 12)
, -
!
,
= (1 2)(3 5)
,
= (1 12)(2 10)(3 8)(4 6)(5 11)(7 9)
, -
!
,
= (1 2 3)
,
= (1 11 6)(2 5 9)(3 7 12)(4 10 8)
, -
!
,
= (1 2 3 4 5)
,
= (1 6 5 3 9)(4 12 7 8 11)
, -
!
,
= (1 2 3 5 4)
,
= (1 4 8 6 2)(5 7 10 12 9)
, -
!
,
= (1 2 4 5 3)
,
= (1 8 7 3 10)(2 12 5 6 11)
, -
!
,
= (1 2 4)
,
= (1 7 4)(2 11 8)(3 5 10)(6 9 12)
, -
!
,
= (1 2 4 3 5)
,
= (1 2 9 11 7)(3 6 12 10 4)
, -
!
,
= (1 2 5 4 3)
,
= (2 3 4 7 5)(6 8 10 11 9)
, -
!
,
= (1 2 5)
,
= (1 9 8)(2 6 3)(4 5 12)(7 11 10)
, -
!
,
= (1 2 5 3 4)
,
= (1 10 5 4 11)(2 8 9 3 12)
, -
!
,
= (1 3 2)
,
= (1 6 11)(2 9 5)(3 12 7)(4 8 10)
, -
!
,
= (1 3 4 5 2)
,
= (2 5 7 4 3)(6 9 11 10 8)
, -
!
,
= (1 3 5 4 2)
,
= (1 10 3 7 8)(2 11 6 5 12)
, -
!
,
= (1 3)(4 5)
,
= (1 7)(2 10)(3 11)(4 5)(6 12)(8 9)
, -
!
,
= (1 3 4)
,
= (1 9 10)(2 12 4)(3 6 8)(5 11 7)
, -
!
,
= (1 3 5)
,
= (1 3 4)(2 8 7)(5 6 10)(9 12 11)
, -
!
,
= (1 3)(2 4)
,
= (1 12)(2 6)(3 9)(4 11)(5 8)(7 10)
, -
!
,
= (1 3 2 4 5)
,
= (1 4 10 11 5)(2 3 8 12 9)
, -
!
,
= (1 3 5 2 4)
,
= (1 5 9 6 3)(4 7 11 12 8)
, -
!
,
= (1 3)(2 5)
,
= (1 2)(3 5)(4 9)(6 7)(8 11)(10 12)
, -
!
,
= (1 3 2 5 4)
,
= (1 11 2 7 9)(3 10 6 4 12)
, -
!
,
= (1 3 4 2 5)
,
= (1 8 2 4 6)(5 10 9 7 12)
, -
!
,
= (1 4 5 3 2)
,
= (1 2 6 8 4)(5 9 12 10 7)
, -
!
,
= (1 4 2)
,
= (1 4 7)(2 8 11)(3 10 5)(6 12 9)
, -
!
,
= (1 4 3 5 2)
,
= (1 11 4 5 10)(2 12 3 9 8)
, -
!
,
= (1 4 3)
,
= (1 10 9)(2 4 12)(3 8 6)(5 7 11)
, -
!
,
= (1 4 5)
,
= (1 5 2)(3 7 9)(4 11 6)(8 10 12)
, -
!
,
= (1 4)(3 5)
,
= (1 6)(2 3)(4 9)(5 8)(7 12)(10 11)
, -
!
,
= (1 4 5 2 3)
,
= (1 9 7 2 11)(3 12 4 6 10)
, -
!
,
= (1 4)(2 3)
,
= (1 8)(2 10)(3 4)(5 12)(6 7)(9 11)
, -
!
,
= (1 4 2 3 5)
,
= (2 7 3 5 4)(6 11 8 9 10)
, -
!
,
= (1 4 2 5 3)
,
= (1 3 6 9 5)(4 8 12 11 7)
, -
!
,
= (1 4 3 2 5)
,
= (1 7 10 8 3)(2 5 11 12 6)
, -
!
,
= (1 4)(2 5)
,
= (1 12)(2 9)(3 11)(4 10)(5 6)(7 8)
, -
!
,
= (1 5 4 3 2)
,
= (1 9 3 5 6)(4 11 8 7 12)
, -
!
,
= (1 5 2)
,
= (1 8 9)(2 3 6)(4 12 5)(7 10 11)
, -
!
,
= (1 5 3 4 2)
,
= (1 7 11 9 2)(3 4 10 12 6)
, -
!
,
= (1 5 3)
,
= (1 4 3)(2 7 8)(5 10 6)(9 11 12)
, -
!
,
= (1 5 4)
,
= (1 2 5)(3 9 7)(4 6 11)(8 12 10)
, -
!
,
= (1 5)(3 4)
,
= (1 12)(2 11)(3 10)(4 8)(5 9)(6 7)
, -
!
,
= (1 5 4 2 3)
,
= (1 5 11 10 4)(2 9 12 8 3)
, -
!
,
= (1 5)(2 3)
,
= (1 10)(2 12)(3 11)(4 7)(5 8)(6 9)
, -
!
,
= (1 5 2 3 4)
,
= (1 3 8 10 7)(2 6 12 11 5)
, -
!
,
= (1 5 2 4 3)
,
= (1 6 4 2 8)(5 12 7 9 10)
, -
!
,
= (1 5 3 2 4)
,
= (2 4 5 3 7)(6 10 9 8 11)
, -
!
,
= (1 5)(2 4)
,
= (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 grou ...
on 5 elements
* ''I
h'', 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 sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
s (the latter of which does not split) and product
:
:
:
In words,
*
is a ''
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
'' of
*
is a ''factor'' of
, which is a ''
direct product''
*
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 ex ...
'' of
Note that
has an
exceptional irreducible 3-dimensional
representation (as the icosahedral rotation group), but
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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
with five elements, which exhibit the subgroups and covering groups directly; none of these are the full icosahedral group:
*
the
projective special linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
, see
here for a proof;
*
the
projective general linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associa ...
;
*
the
special linear group.
Conjugacy classes
The 120 symmetries fall into 10 conjugacy classes.
{, class=wikitable
, +
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
es
!''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".
{, class="wikitable sortable"
!
Schön., , colspan=2,
Coxeter, ,
Orb., ,
H-M, ,
Structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
, ,
Cyc., ,
Order, ,
Index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
, , Mult., , Description
, - align=center BGCOLOR="#e0f0f0"
, I
h, ,
,3, , , *532, , 2/m, ,
A5×Z
2, , , , 120, , 1, , 1, , full group
, - align=center BGCOLOR="#e0f0f0"
, D
2h, ,
,2, , , *222, , mmm, ,
D4×D
2=D
23, ,

, , 8, , 15, , 5, , fixing two opposite edges, possibly swapping them
, -align=center BGCOLOR="#e0f0f0"
, C
5v , ,
, , , *55 , , 5m, , D
10, ,

, , 10 , , 12, , 6, , fixing a face
, -align=center BGCOLOR="#e0f0f0"
, C
3v , ,
, , , *33 , , 3m, , D
6=S
3, ,

, , 6 , , 20, , 10, , fixing a vertex
, -align=center BGCOLOR="#e0f0f0"
, C
2v , ,
, , , *22 , , 2mm, , D
4=D
22, ,

, , 4 , , 30, , 15, , fixing an edge
, -align=center BGCOLOR="#e0f0f0"
, C
s , ,
nbsp;, , , * , , or m, , D
2, ,

, , 2 , , 60, , 15, , reflection swapping two endpoints of an edge
, - align=center BGCOLOR="#f0f0e0"
, T
h, ,
+,4">+,4, , , 3*2, , m, , A
4×Z
2, ,

, , 24, , 5, , 5, , pyritohedral group
, -align=center BGCOLOR="#f0f0e0"
, D
5d , ,
+,10">+,10, , , 2*5 , , m2, , D
20=Z
2×D
10, ,

, , 20 , , 6, , 6, , fixing two opposite faces, possibly swapping them
, -align=center BGCOLOR="#f0f0e0"
, D
3d , ,
+,6">+,6, , , 2*3 , , m, , D
12=Z
2×D
6, ,

, , 12 , , 10, , 10, , fixing two opposite vertices, possibly swapping them
, -align=center BGCOLOR="#f0f0e0"
, D
1d = C
2h , ,
+,2">+,2, , , 2* , , 2/m, , D
4=
Z2×D
2, ,

, , 4 , , 30, , 15, , halfturn around edge midpoint, plus central inversion
, -align=center BGCOLOR="#e0e0e0"
, S
10 , ,
+,10+">+,10+, , , 5× , , , , Z
10=Z
2×Z
5, ,

, , 10 , , 12, , 6, , rotations of a face, plus central inversion
, -align=center BGCOLOR="#e0e0e0"
, S
6 , ,
+,6+">+,6+, , , 3× , , , , Z
6=Z
2×Z
3, ,

, , 6 , , 20, , 10, , rotations about a vertex, plus central inversion
, -align=center BGCOLOR="#e0e0e0"
, S
2 , ,
+,2+">+,2+, , , × , , , , Z
2, ,

, , 2 , , 60, , 1, , central inversion
, -align=center BGCOLOR="#f0e0f0"
, I, ,
,3sup>+, , , , 532, , 532, , A
5, , , , 60, , 2, , 1, , all rotations
, - align=center BGCOLOR="#f0e0f0"
, T, ,
,3sup>+, , , , 332, , 332, , A
4 , ,

, , 12, , 10, , 5, , rotations of a contained tetrahedron
, - align=center BGCOLOR="#f0e0f0"
, D
5, ,
,5sup>+, , , , 522, , 522, , D
10, ,

, , 10, , 12, , 6, , rotations around the center of a face, and h.t.s.(face)
, - align=center BGCOLOR="#f0e0f0"
, D
3, ,
,3sup>+, , , , 322, , 322, , D
6=S
3, ,

, , 6, , 20, , 10, , rotations around a vertex, and h.t.s.(vertex)
, - align=center BGCOLOR="#f0e0f0"
, D
2, ,
,2sup>+, , , , 222, , 222, , D
4=Z
22, ,

, , 4, , 30, , 5, , halfturn around edge midpoint, and h.t.s.(edge)
, - align=center BGCOLOR="#f0e0f0"
, C
5, ,
sup>+, , , , 55, , 5, , Z
5, ,

, , 5, , 24, , 6, , rotations around a face center
, - align=center BGCOLOR="#f0e0f0"
, C
3, ,
sup>+, , , , 33, , 3, , Z
3=A
3, ,

, , 3, , 40, , 10, , rotations around a vertex
, - align=center BGCOLOR="#f0e0f0"
, C
2, ,
sup>+, , , , 22, , 2, , Z
2, ,

, , 2, , 60, , 15, , half-turn around edge midpoint
, - align=center BGCOLOR="#f0e0f0"
, C
1, ,
nbsp;sup>+, , , , 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 group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
s ''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
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-group
In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...
s
* stabilizers of a pair of edges in ''I'' give
Klein four-group
In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...
s
; there are 5 of these, given by rotation by 180° in 3 perpendicular axes.
* stabilizers of a pair of edges in ''I
h'' give
; 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
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
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
Polyhedron stabilizers
For each of these, there are 5 conjugate copies, and the conjugation action gives a map, indeed an isomorphism,
.
* 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
,3() of order 120 has generators represented by the reflection matrices R
0, R
1, R
2 below, with relations R
02 = R
12 = R
22 = (R
0×R
1)
5 = (R
1×R
2)
3 = (R
0×R
2)
2 = Identity. The group
,3sup>+ () 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
denotes the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
.
{, class=wikitable
, +
,3
!
!colspan=3, Reflections
!colspan=3, Rotations
!Rotoreflection
, -
!Name
! R
0
! R
1
! R
2
! S
0,1
! S
1,2
! S
0,2
! V
0,1,2
, - align=center
!Group
,
,
,
,
,
,
,
, - align=center
!Order
, 2, , 2, , 2, , 5, , 3, , 2, , 10
, - align=center
!Matrix
,