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: 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 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

Main ATLAS page Go to main ATLAS (version 2.0) page.
Classical groups page Go to classical groups page.
Old O8+(3) page Go to old O8+(3) page - ATLAS version 1.
ftp access 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.