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

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.

Automorphisms

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)).

Representations

The representations of O8+(3) available are:
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:
The representation of 2^2.O8+(3).S4 available is:

Maximal subgroups

The maximal subgroups of O8+(3) are as follows:

Checks applied

CheckDateBy whomRemarks
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.