<s>
In	O
mathematics	O
,	O
a	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
is	O
an	O
important	O
type	O
of	O
group	O
that	O
can	O
be	O
realized	O
as	O
the	O
group	O
of	O
symmetries	O
of	O
a	O
hypercube	B-Operating_System
or	O
of	O
a	O
cross-polytope	O
.	O
</s>
<s>
Groups	O
of	O
this	O
type	O
are	O
identified	O
by	O
a	O
parameter	O
,	O
the	O
dimension	O
of	O
the	O
hypercube	B-Operating_System
.	O
</s>
<s>
As	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
it	O
is	O
of	O
type	O
,	O
and	O
as	O
a	O
Weyl	O
group	O
it	O
is	O
associated	O
to	O
the	O
symplectic	O
groups	O
and	O
with	O
the	O
orthogonal	O
groups	O
in	O
odd	O
dimensions	O
.	O
</s>
<s>
As	O
a	O
wreath	O
product	O
it	O
is	O
where	O
is	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
of	O
degree	O
.	O
</s>
<s>
As	O
a	O
permutation	B-Algorithm
group	I-Algorithm
,	O
the	O
group	O
is	O
the	O
signed	B-Algorithm
symmetric	I-Algorithm
group	I-Algorithm
of	O
permutationsπ	O
either	O
of	O
the	O
set	O
or	O
of	O
the	O
set	O
such	O
that	O
for	O
all	O
.	O
</s>
<s>
As	O
a	O
matrix	B-Algorithm
group	I-Algorithm
,	O
it	O
can	O
be	O
described	O
as	O
the	O
group	O
of	O
orthogonal	B-Algorithm
matrices	I-Algorithm
whose	O
entries	O
are	O
all	O
integers	O
.	O
</s>
<s>
The	O
representation	O
theory	O
of	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
was	O
described	O
by	O
according	O
to	O
.	O
</s>
<s>
In	O
three	O
dimensions	O
,	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
is	O
known	O
as	O
where	O
is	O
the	O
octahedral	O
group	O
,	O
and	O
is	O
a	O
symmetric	B-Algorithm
group	I-Algorithm
(	O
here	O
a	O
cyclic	O
group	O
)	O
of	O
order	O
2	O
.	O
</s>
<s>
In	O
two	O
dimensions	O
,	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
structure	O
is	O
the	O
abstract	O
dihedral	B-Algorithm
group	I-Algorithm
of	O
order	O
eight	O
,	O
describing	O
the	O
symmetry	O
of	O
a	O
square	O
,	O
or	O
2-orthoplex	O
.	O
</s>
<s>
Hyperoctahedral	B-Algorithm
groups	I-Algorithm
can	O
be	O
named	O
as	O
Bn	O
,	O
a	O
bracket	O
notation	O
,	O
or	O
as	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
graph	O
:	O
</s>
<s>
=	O
483+6	O
Cube	B-Application
,	O
octahedron4±1/6[OxO].2	O
(	O
O/V	O
;O	O
/V	O
)	O
*	O
B4[4,3,3]244	O
!	O
</s>
<s>
=	O
3844+12	O
Tesseract	B-Language
,	O
16-cell	O
,	O
24-cell5	O
B5[4,3,3,3]255	O
!	O
</s>
<s>
=	O
38405+20	O
5-cube	O
,	O
5-orthoplex6	O
B6[4,34]266	O
!	O
</s>
<s>
=	O
460806+30	O
6-cube	O
,	O
orthoplex	O
...	O
n	O
2nn	O
...	O
=	O
(	O
2n	O
)	O
!	O
</s>
<s>
There	O
is	O
a	O
notable	O
index	O
two	O
subgroup	O
,	O
corresponding	O
to	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
Dn	O
and	O
the	O
symmetries	O
of	O
the	O
demihypercube	O
.	O
</s>
<s>
Viewed	O
as	O
a	O
wreath	O
product	O
,	O
there	O
are	O
two	O
natural	O
maps	O
from	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
to	O
the	O
cyclic	O
group	O
of	O
order	O
2	O
:	O
one	O
map	O
coming	O
from	O
"	O
multiply	O
the	O
signs	O
of	O
all	O
the	O
elements	O
"	O
(	O
in	O
the	O
n	O
copies	O
of	O
)	O
,	O
and	O
one	O
map	O
coming	O
from	O
the	O
parity	O
of	O
the	O
permutation	O
.	O
</s>
<s>
The	O
kernel	O
of	O
the	O
first	O
map	O
is	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
In	O
terms	O
of	O
signed	O
permutations	O
,	O
thought	O
of	O
as	O
matrices	O
,	O
this	O
third	O
map	O
is	O
simply	O
the	O
determinant	O
,	O
while	O
the	O
first	O
two	O
correspond	O
to	O
"	O
multiplying	O
the	O
non-zero	O
entries	O
"	O
and	O
"	O
parity	O
of	O
the	O
underlying	O
(	O
unsigned	O
)	O
permutation	O
"	O
,	O
which	O
are	O
not	O
generally	O
meaningful	O
for	O
matrices	O
,	O
but	O
are	O
in	O
the	O
case	O
due	O
to	O
the	O
coincidence	O
with	O
a	O
wreath	O
product	O
.	O
</s>
<s>
The	O
kernels	O
of	O
these	O
three	O
maps	O
are	O
all	O
three	O
index	O
two	O
subgroups	O
of	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
,	O
as	O
discussed	O
in	O
H1	O
:	O
Abelianization	O
below	O
,	O
and	O
their	O
intersection	O
is	O
the	O
derived	O
subgroup	O
,	O
of	O
index	O
4	O
(	O
quotient	O
the	O
Klein	O
4-group	O
)	O
,	O
which	O
corresponds	O
to	O
the	O
rotational	O
symmetries	O
of	O
the	O
demihypercube	O
.	O
</s>
<s>
In	O
dimension	O
2	O
these	O
groups	O
completely	O
describe	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
,	O
which	O
is	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
Dih4	O
of	O
order	O
8	O
,	O
and	O
is	O
an	O
extension	O
2.V	O
(	O
of	O
the	O
4-group	O
by	O
a	O
cyclic	O
group	O
of	O
order	O
2	O
)	O
.	O
</s>
<s>
The	O
group	O
homology	O
of	O
the	O
hyperoctahedral	B-Algorithm
group	I-Algorithm
is	O
similar	O
to	O
that	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
and	O
exhibits	O
stabilization	O
,	O
in	O
the	O
sense	O
of	O
stable	O
homotopy	O
theory	O
.	O
</s>
