ATLAS: Orthogonal group O8+(3)
Order = 4952179814400 = 212.312.52.7.13.
Mult = 2 × 2.
Out = S4.
The following information is available for O8+(3):
Standard generators of O8+(3) are
a
and b where
a is in class 2A,
b is in class 5B,
ab has order 13,
and abb has order 14.
Standard generators of 2.O8+(3) are preimages A and B where
A has order 2, B has order 5 and AB has order 13.
Standard generators of 2^2.O8+(3) are preimages A and B where
B has order 5 and AB has order 13.
NB: Since classes 2A/B/C and classes 5A/B/C are automorphic, an alternative
definition of standard generators is as follows: a and b where
a is in class 2A/B/C, b is in class 5A/B/C, ab has
order 13 and abb has order 14. It is not possible for these properties
to be satisfied with a in class 2X and b in class
5X for any X in {A,B,C}. Wlog, we can label the classes
so that a is in class 2A and b is in class 5B.
An outer automorphism swapping classes 5B and 5C may be obtained by
mapping (a, b) to (a, (((ab)^4abbababb)^4)^((abb)^5)).
An outer automorphism swapping classes 5A and 5C may be obtained by
mapping (a, b) to (((ab)^4abbababb)^10, b^((abb)^6)).
G2 generators for O8+(3):S4 are the above two automorphisms.
They have orders 24 and 20 respectively, and may be taken to map to
permutations (1423) and (1234) of the outer S4.
An outer automorphism acting as (12)(34) is (a, b) -> (a, b^-1).
An outer automorphism acting as (12) is (a, b) -> (a, ((ababab^4)^4)^(b^2)).
An outer automorphism acting as (234) is (a, b) -> ((abb)^7, ((abab^4ab^4)^4)^(ab^4ab^4)).
The representations of O8+(3) available are:
-
Permutations on 1080a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1080b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1080c points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1080d points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1080e points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1080f points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1120a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1120b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 1120c points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 28431a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 28431b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 28431c points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 28431d points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 36400 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 298 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 28 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 35a over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 35b over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 35c over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 195 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 322 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 518a over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 518b over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 518c over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 567a over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 567b over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 567c over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 300 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 299 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 300 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.O8+(3) available are:
-
Permutations on 2160a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 2160b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 2240 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).
-
Dimension 56 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 104 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 224a over GF(3) - in 8 × 35b:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 224b over GF(3) - in 8 × 35c:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 384 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representation of 2.O8+(3):2_1 available is
The representation of 2.O8+(3):2_2 available is
The representations of O8+(3).S4 available are:
-
Permutations on 3360 points:
x and
y (Meataxe),
x and
y (Meataxe binary),
x and
y (GAP).
-
Permutations on 6480 points:
x and
y (Meataxe),
x and
y (Meataxe binary),
x and
y (GAP).
The representation of 2^2.O8+(3).S4 available is:
-
Dimension 24 over GF(3):
X and
Y (Meataxe),
X and
Y (Meataxe binary),
X and
Y (GAP).
The maximal subgroups of O8+(3) are as follows:
-
O7(3), with generators
a, b^2ab^2.
-
O7(3), with generators
a, bab^2ab^2ab^4.
-
O7(3).
-
O7(3).
-
O7(3).
-
O7(3).
-
3^6:L4(3), with generators
a, abab^3.
-
3^6:L4(3).
-
3^6:L4(3).
-
O8+(2), with generators
a, abababab^2ab.
-
O8+(2), with generators
a, abab^2ababab.
-
O8+(2), with generators
a, abab^3abab^2ab.
-
O8+(2), with generators
a, abab^2abab^3ab.
-
3^{1+8}:2(A4 × A4 × A4).2.
-
2.U4(3).2^2.
-
2.U4(3).2^2.
-
2.U4(3).2^2.
-
(A4 × U4(2)):2.
-
(A4 × U4(2)):2.
-
(A4 × U4(2)):2.
-
(A4 × U4(2)):2.
-
(A4 × U4(2)):2.
-
(A4 × U4(2)):2.
-
(A6 × A6):2^2.
-
(A6 × A6):2^2.
-
(A6 × A6):2^2.
-
2^{1+8}.3^4.2^3 = 2.(A4 wr 2^2).2.
| 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 | | |
| All representations are in standard generators | |
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old O8+(3) 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 3rd February 2003.
Last updated 06.02.03 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.