ATLAS: Orthogonal group O_{7}(3)
Order = 4585351680 = 2^{9}.3^{9}.5.7.13.
Mult = 6.
Out = 2.
The following information is available for O7(3):
Standard generators of O_{7}(3) are a and b where
a is in class 2A, b has order 7, ab has order 13
and abb has order 20.
Standard generators of the double cover 2.O_{7}(3) are preimages
A and B where B has order 7 and AB has order 13.
Standard generators of the triple cover 3.O_{7}(3) are preimages
A and B where A has order 2 and B has order 7.
Standard generators of the sextuple cover 6.O_{7}(3) are preimages
A and B where
A has order 4, B has order 7,
and AB has order 39.
Standard generators of O_{7}(3):2 are
c
and d where
c is in class 2D,
d has order 7,
and cd has order 26,
and cdcdd has order 14.
Standard generators of the double cover 2.O_{7}(3):2 are preimages
C
and D where
D has order 7.
Standard generators of the triple cover 3O_{7}(3):2 are preimages
C
and D where
D has order 7.
Standard generators of the sixfold cover 6.O_{7}(3):2 are preimages
C and D where
D has order 7.
The outer automorphism of O_{7}(3) may be realised by mapping
(a, b) to (a, b^{1}). This automorphism resides in
Class 2F.
To find standard generators for O_{7}(3):
 Find an element of order 14. This powers up to x in class 2A and
y of order 7.
 Find a conjugate a of x and a conjugate b of y
such that ab has order 13 and abb has order 20.
To find standard generators for O_{7}(3).2:
 Find an element of order 26. This powers up to x in class 2D
 Find an element of order 7, 14 or 28. This powers up to
y of order 7.
 Find a conjugate c of x and a conjugate d of y
such that cd has order 26 and cdcdd has order 14.
The representations of O7(3) available are
 Some primitive permutation representations.

Permutations on 351 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 364 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 378 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1080 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1080 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1120 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 3640 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some faithful irreducibles in characteristic 2.

Dimension 78 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 90 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 104 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 260 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 260 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some faithful irreducibles in characteristic 3.

Dimension 7 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 21 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 63 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 189 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 309 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.O7(3) available are

Permutations on 2160 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 representation of 3.O7(3) available is:

Dimension 27 over GF(4):
A and
B' (Meataxe),
A and
B' (Meataxe binary),
A and
B' (GAP).
— WRONG generators. Correct (standard) generators are (A, B'B').
The representations of O7(3):2 available are:

Permutations on 351 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 78 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 7 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representation of 2.O7(3).2 available is

Dimension 8 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representation of 3.O7(3):2 available is

Dimension 54 over GF(2):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of O_{7}(3) are as follows.

2U4(3).2
 3^5:U4(2):2
 L4(3):2
 G2(3)
 G2(3)
 3^3+3:L3(3)
 S6(2)
 S6(2)
 3^1+6:(2A4 x A4).2
 S9
 S9
 (2^2 x U4(2)):2
 2^6:A7
 S4 x S6
 S4 x 2(A4 x A4).2
The maximal subgroups of O_{7}(3):2 are as follows.
 O7(3)

2U4(3).2.2
 3^5:(U4(2):2 x 2)
 L4(3):2 x 2
 3^3+3:(L3(3) x 2)
 3^1+6:(2S4 x S4)
 D8 x U4(2):2
 2^6:S7
 S4 x S6 x 2
 S4 x 2(A4 x A4).4
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  24.01.01  RAW 
All representations are in standard generators  
Go to main ATLAS (version 2.0) page.
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See here for details.
Version 2.0 created on 24th January 2001.
Last updated 25.01.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.