<s>
Every	O
polyhedron	O
with	O
icosahedral	O
symmetry	O
has	O
60	O
rotational	O
(	O
or	O
orientation-preserving	O
)	O
symmetries	O
and	O
60	O
orientation-reversing	O
symmetries	O
(	O
that	O
combine	O
a	O
rotation	O
and	O
a	O
reflection	B-Algorithm
)	O
,	O
for	O
a	O
total	O
symmetry	O
order	O
of	O
120	O
.	O
</s>
<s>
The	O
full	O
symmetry	O
group	O
is	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
of	O
type	O
.	O
</s>
<s>
The	O
set	O
of	O
rotational	O
symmetries	O
forms	O
a	O
subgroup	O
that	O
is	O
isomorphic	O
to	O
the	O
alternating	B-Algorithm
group	I-Algorithm
on	O
5	O
letters	O
.	O
</s>
<s>
Apart	O
from	O
the	O
two	O
infinite	O
series	O
of	O
prismatic	O
and	O
antiprismatic	O
symmetry	O
,	O
rotational	O
icosahedral	O
symmetry	O
or	O
chiral	O
icosahedral	O
symmetry	O
of	O
chiral	O
objects	O
and	O
full	O
icosahedral	O
symmetry	O
or	O
achiral	O
icosahedral	O
symmetry	O
are	O
the	O
discrete	B-Algorithm
point	I-Algorithm
symmetries	I-Algorithm
(	O
or	O
equivalently	O
,	O
symmetries	O
on	O
the	O
sphere	O
)	O
with	O
the	O
largest	O
symmetry	O
groups	O
.	O
</s>
<s>
Note	O
that	O
other	O
presentations	O
are	O
possible	O
,	O
for	O
instance	O
as	O
an	O
alternating	B-Algorithm
group	I-Algorithm
(	O
for	O
I	O
)	O
.	O
</s>
<s>
The	O
full	O
symmetry	O
group	O
is	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
of	O
type	O
.	O
</s>
<s>
The	O
set	O
of	O
rotational	O
symmetries	O
forms	O
a	O
subgroup	O
that	O
is	O
isomorphic	O
to	O
the	O
alternating	B-Algorithm
group	I-Algorithm
on	O
5	O
letters	O
.	O
</s>
<s>
Every	O
polyhedron	O
with	O
icosahedral	O
symmetry	O
has	O
60	O
rotational	O
(	O
or	O
orientation-preserving	O
)	O
symmetries	O
and	O
60	O
orientation-reversing	O
symmetries	O
(	O
that	O
combine	O
a	O
rotation	O
and	O
a	O
reflection	B-Algorithm
)	O
,	O
for	O
a	O
total	O
symmetry	O
order	O
of	O
120	O
.	O
</s>
<s>
Each	O
octahedron	O
can	O
represent	O
3	O
orthogonal	O
mirror	O
planes	O
by	O
its	O
edges.160px200pxThe	O
pyritohedral	O
symmetry	O
is	O
an	O
index	O
5	O
subgroup	O
of	O
icosahedral	O
symmetry	O
,	O
with	O
3	O
orthogonal	O
green	O
reflection	B-Algorithm
lines	O
and	O
8	O
red	O
order-3	O
gyration	O
points	O
.	O
</s>
<s>
The	O
group	O
I	O
is	O
isomorphic	O
to	O
A5	O
,	O
the	O
alternating	B-Algorithm
group	I-Algorithm
of	O
even	O
permutations	O
of	O
five	O
objects	O
.	O
</s>
<s>
The	O
group	O
Ih	O
is	O
isomorphic	O
to	O
I	O
×	O
Z2	O
,	O
or	O
A5	O
×	O
Z2	O
,	O
with	O
the	O
inversion	B-Algorithm
in	I-Algorithm
the	I-Algorithm
center	I-Algorithm
corresponding	O
to	O
element	O
(	O
identity	O
,	O
-1	O
)	O
,	O
where	O
Z2	O
is	O
written	O
multiplicatively	O
.	O
</s>
<s>
Ih	O
acts	O
on	O
the	O
compound	O
of	O
five	O
cubes	O
and	O
the	O
compound	O
of	O
five	O
octahedra	O
,	O
but	O
−1	O
acts	O
as	O
the	O
identity	O
(	O
as	O
cubes	O
and	O
octahedra	O
are	O
centrally	B-Algorithm
symmetric	I-Algorithm
)	O
.	O
</s>
<s>
Since	O
the	O
permutations	O
Pi	O
are	O
all	O
the	O
60	O
even	O
permutations	O
of	O
12345	O
,	O
the	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
is	O
made	O
explicit	O
,	O
therefore	O
the	O
isomorphism	O
too	O
.	O
</s>
<s>
Note	O
that	O
has	O
an	O
exceptional	O
irreducible	O
3-dimensional	O
representation	O
(	O
as	O
the	O
icosahedral	O
rotation	O
group	O
)	O
,	O
but	O
does	O
not	O
have	O
an	O
irreducible	O
3-dimensional	O
representation	O
,	O
corresponding	O
to	O
the	O
full	B-Algorithm
icosahedral	I-Algorithm
group	I-Algorithm
not	O
being	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
These	O
can	O
also	O
be	O
related	O
to	O
linear	O
groups	O
over	O
the	O
finite	O
field	O
with	O
five	O
elements	O
,	O
which	O
exhibit	O
the	O
subgroups	O
and	O
covering	O
groups	O
directly	O
;	O
none	O
of	O
these	O
are	O
the	O
full	B-Algorithm
icosahedral	I-Algorithm
group	I-Algorithm
:	O
</s>
<s>
The	O
full	O
icosahedral	O
symmetry	O
group	O
 [ 5 , 3 ] 	O
(	O
)	O
of	O
order	O
120	O
has	O
generators	O
represented	O
by	O
the	O
reflection	B-Algorithm
matrices	O
R0	O
,	O
R1	O
,	O
R2	O
below	O
,	O
with	O
relations	O
R02	O
=	O
R12	O
=	O
R22	O
=	O
(	O
R0×R1	O
)	O
5	O
=	O
(	O
R1×R2	O
)	O
3	O
=	O
(	O
R0×R2	O
)	O
2	O
=	O
Identity	O
.	O
</s>
<s>
A	O
rotoreflection	B-Algorithm
of	O
order	O
10	O
is	O
generated	O
by	O
V0	O
,	O
1	O
,	O
2	O
,	O
the	O
product	O
of	O
all	O
3	O
reflections	O
.	O
</s>
<s>
Fundamental	O
domains	O
for	O
the	O
icosahedral	O
rotation	O
group	O
and	O
the	O
full	B-Algorithm
icosahedral	I-Algorithm
group	I-Algorithm
are	O
given	O
by	O
:	O
</s>
