ATLAS: Orthogonal group O7(3)
Order = 4585351680 = 29.39.5.7.13.
Mult = 6.
Out = 2.
The following information is available for O7(3):
Standard generators of O7(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.O7(3) are preimages
A and B where B has order 7 and AB has order 13.
Standard generators of the triple cover 3.O7(3) are preimages
A and B where A has order 2 and B has order 7.
Standard generators of the sextuple cover 6.O7(3) are preimages
A and B where
A has order 4, B has order 7,
and AB has order 39.
Standard generators of O7(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.O7(3):2 are preimages
C
and D where
D has order 7.
Standard generators of the triple cover 3O7(3):2 are preimages
C
and D where
D has order 7.
Standard generators of the sixfold cover 6.O7(3):2 are preimages
C and D where
D has order 7.
The outer automorphism of O7(3) may be realised by mapping
(a, b) to (a, b-1). This automorphism resides in
Class 2F.
To find standard generators for O7(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 O7(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 O7(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 O7(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.
Go to classical groups page.
Go to old O7(3) page - ATLAS version 1.
Anonymous ftp access is also available.
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.