ATLAS: Orthogonal group O_{8}^{}(3)
Order = 10151968619520 = 2^{10}.3^{12}.5.7.13.41.
Mult = 2.
Out = 2 × 2.
The following information is available for O_{8}^{}(3):
Standard generators of O_{8}^{}(3) are a and b where
a is in class 2A, b is in class 4F, ab has order 41,
abb has order 6 and ababb has order 41.
Standard generators of the double cover 2.O_{8}^{}(3) are preimages A
and B where AB has order 41 and ABABB has order 41.
Standard generators of O_{8}^{}(3):2_{1} are
c and d where c is in class 2D/E, d is in class 8F, cd has order 41, cdd has order 12 and cdcdd has order 18.
We may obtain d as d = cx, where x has order 41, cx has order 8, cxx has order 12 and cxxx has order 18.
Standard generators of the double cover
2.O_{8}^{}(3):2_{1} are preimages
C and D where C has order 2 and CD has order 41.
Type I standard generators of O_{8}^{}(3).2_{3} are
g and h where g is in class 3A, h is in class 8V/W, gh has order 56 and ghh has order 36.
Type II standard generators of O_{8}^{}(3).2_{3} are
g_{2} and h_{2} where g_{2} is
in class 2A, h_{2} is in class 8X/Y,
g_{2}h_{2} has order 24,
g_{2}h_{2}h_{2} has order 7,
g_{2}h_{2}h_{2}h_{2} has order 56 and
g_{2}h_{2}h_{2}h_{2}h_{2} has order 3.
Standard generators of O_{8}^{}(3).2^{2} are
i and j where j is in class 2DE, j is in class 8B1, ij has order 82 and ijj has order 8.
We may obtain j as j = ix, where x has order 82, ix has order 8 and ixx has order 8.
Note 1: Some class definitions are here.
Note 2: SO_{8}^{}(3) = O_{8}^{}(3) × 2 and
GO_{8}^{}(3) = O_{8}^{}(3):2_{1} × 2.
Note 3: There are no double covers of O_{8}^{}(3).2_{3}
and O_{8}^{}(3).2^{2} [that contain
2.O_{8}^{}(3)].
Note 4: No standard generators were defined in v1. Any representations in v1
will be on different generators.
The representations of O_{8}^{}(3) available are:

Permutations on 1066 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.
The representations of 2.O_{8}^{}(3) available are:

Dimension 8 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 the spin representation.

Dimension 8 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 the Frobenius automorph of the above.

Dimension 16 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).
The representations of O_{8}^{}(3):2_{1} available are:

Permutations on 1066 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 8 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 the natural representation.
The representations of 2.O_{8}^{}(3):2_{1} available are:

Dimension 16 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of O_{8}^{}(3).2_{3} available are:

Permutations on 1066 points:
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).

Permutations on 1066 points:
g_{2} and
h_{2} (Meataxe),
g_{2} and
h_{2} (Meataxe binary),
g_{2} and
h_{2} (GAP).

Dimension 8 over GF(3):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
The representations of O_{8}^{}(3).2^{2} available are:

Permutations on 1066 points:
i and
j (Meataxe),
i and
j (Meataxe binary),
i and
j (GAP).

Dimension 16 over GF(3):
i and
j (Meataxe),
i and
j (Meataxe binary),
i and
j (GAP).

Dimension 8 over GF(3):
i and
j (Meataxe),
i and
j (Meataxe binary),
i and
j (GAP).
 the matrices for this last group generate O_{8}^{}(3).D_{8} [a double cover of
O_{8}^{}(3).2^{2}!].
This group contains copies of
O_{8}^{}(3):2_{1} and O_{8}^{}(3).2_{3}, but does not contain
O_{8}^{}(3):2_{2}.
The maximal subgroups of O_{8}^{}(3) are as follows  INCOMPLETE:
The maximal subgroups of O_{8}^{}(3):2_{1} are as follows  INCOMPLETE:
The maximal subgroups of O_{8}^{}(3):2_{2} are as follows  INCOMPLETE:
The maximal subgroups of O_{8}^{}(3).2_{3} are as follows  INCOMPLETE:
The maximal subgroups of O_{8}^{}(3).2^{2} are as follows  INCOMPLETE:
Note: The L_{2}(81) normalisers have abstract specification
C(2H). The outer automorphisms of L_{2}(81):2_{1} and
L_{2}(81):4_{1} are field automorphisms; this
is for consistency with the GAP tables.
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Go to old O8(3) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 24th August 2000.
Last updated 23.02.04 by JNB.
Information checked to
Level 0 on 24.08.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.