ATLAS: Orthogonal group O8(2)
Order = 197406720.
Mult = 1.
Out = 2.
See also ATLAS of Finite Groups, pp8889.
The following information is available for O8(2):
Standard generators of O8(2) are
a
and b where
a is in class 2C,
b is in class 3C
and ab has order 17,
ababb has order 17,
and abababb has order 30.
Standard generators of O8(2):2 are
c
and d where
c is in class 2D,
d has order 14,
and cd has order 17,
and cdcdd has order 12.
Representations
The representations of O8(2) available are
 Some primitive permutation representations.

Permutations on 119 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 136 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 765 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1071 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1632 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 24192 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 45696 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 2.

Dimension 8 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 26 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 48 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 48 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 48 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 3.

Dimension 34 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 50 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 5.

Dimension 34 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 51 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 84 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 7.

Dimension 33 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 51 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 84 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations in characteristic 17.

Dimension 34 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 51 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 83 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some faithful irreducibles in characteristic 0
 Dimension 34 over Z:
a and b (GAP).
 Dimension 51 over Z:
a and b (GAP).
 Dimension 84 over Z:
a and b (GAP).
 Dimension 204[b] over Z:
a and b (GAP).
The representations of O8(2):2 available are

Dimension 8 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 34 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 50 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 119 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
Maximal subgroups
The maximal subgroups of O8(2) are as follows.
 2^6:U4(2)
 S6(2)
 2^(3+6):(L3(2) x 3)
 2^(1+8):(S3 x A5)
 (3 x A8):2
 L2(16):2
 (S3 x S3 x A5):2
 L2(7)
The maximal subgroups of O8(2):2 are as follows.
 O8(2)
 2^6:U4(2):2
 S6(2) x 2
 2^(3+6):(L3(2) x S3)
 2^(1+8):(S3 x S5)
 S3 x S8
 L2(16):4
 (S3 x S3):2 x S5
 L2(7):2
A set of generators for the maximal cyclic subgroups of O8(2) can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
These classes are compatible with the Atlas of Brauer Characters.
A set of generators for the maximal cyclic subgroups of O8(2):2 can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
These classes are compatible with the Atlas of Brauer Characters.
Check  Date  By whom  Remarks 
Links work (except representations)    
Links to (meataxe) representations work and have right degree and field   
All info from v1 is included   
HTML page standard   
Word program syntax   
Word programs applied   
All necessary standard generators are defined  26.01.01  RAW 
All representations are in standard generators  
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old O8(2) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk, user atlasftp, password atlasftp.
Files can be found in directory v2.0 and subdirectories.
Version 2.0 created on 25th January 2001.
Last updated 04.03.04 by SJN.
Information checked to
Level 0 on 25.01.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.