# ATLAS: Orthogonal group O8-(3)

Order = 10151968619520 = 210.312.5.7.13.41.
Mult = 2.
Out = 2 × 2.

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 4F, ab has order 41, abb has order 6 and ababb has order 41.
Standard generators of the double cover 2.O8-(3) are preimages A and B where AB has order 41 and ABABB has order 41.

Standard generators of O8-(3):21 are c and d where c is in class 2D/E, d is in class 8F, cd has order 41, cdd has order 12 and cdcdd has order 18.
We may obtain d as d = cx, where x has order 41, cx has order 8, cxx has order 12 and cxxx has order 18.
Standard generators of the double cover 2.O8-(3):21 are preimages C and D where C has order 2 and CD has order 41.

Type I standard generators of O8-(3).23 are g and h where g is in class 3A, h is in class 8V/W, gh has order 56 and ghh has order 36.
Type II standard generators of O8-(3).23 are g2 and h2 where g2 is in class 2A, h2 is in class 8X/Y, g2h2 has order 24, g2h2h2 has order 7, g2h2h2h2 has order 56 and g2h2h2h2h2 has order 3.

Standard generators of O8-(3).22 are i and j where j is in class 2DE, j is in class 8B1, ij has order 82 and ijj has order 8.
We may obtain j as j = ix, where x has order 82, ix has order 8 and ixx has order 8.

Note 1: Some class definitions are here.
Note 2: SO8-(3) = O8-(3) × 2 and GO8-(3) = O8-(3):21 × 2.
Note 3: There are no double covers of O8-(3).23 and O8-(3).22 [that contain 2.O8-(3)].
Note 4: No standard generators were defined in v1. Any representations in v1 will be on different generators.

### Representations

The representations of O8-(3) available are:
• Permutations on 1066 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 natural representation.
The representations of 2.O8-(3) available are:
• Dimension 8 over GF(9): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - the spin representation.
• Dimension 8 over GF(9): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - the Frobenius automorph of the above.
• Dimension 16 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - reducible over GF(9).
The representations of O8-(3):21 available are:
• Permutations on 1066 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 8 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - the natural representation.
The representations of 2.O8-(3):21 available are:
The representations of O8-(3).23 available are:
The representations of O8-(3).22 available are:
• Permutations on 1066 points: i and j (Meataxe), i and j (Meataxe binary), i and j (GAP).
• Dimension 16 over GF(3): i and j (Meataxe), i and j (Meataxe binary), i and j (GAP).
• Dimension 8 over GF(3): i and j (Meataxe), i and j (Meataxe binary), i and j (GAP).
- the matrices for this last group generate O8-(3).D8 [a double cover of O8-(3).22!].
This group contains copies of O8-(3):21 and O8-(3).23, but does not contain O8-(3):22.

### Maximal subgroups

The maximal subgroups of O8-(3) are as follows - INCOMPLETE:
The maximal subgroups of O8-(3):21 are as follows - INCOMPLETE:
The maximal subgroups of O8-(3):22 are as follows - INCOMPLETE:
The maximal subgroups of O8-(3).23 are as follows - INCOMPLETE:
The maximal subgroups of O8-(3).22 are as follows - INCOMPLETE:
• O8-(3):21.
• O8-(3):22.
• O8-(3).23.
• 36:2.U4(3).D8.
• 33+6:(L3(3) × SD16).
• 31+8:(2S4 × A6.2^2) [???].
• L4(3)..
• L2(81):41 × 2.

Note: The L2(81) normalisers have abstract specification C(2H). The outer automorphisms of L2(81):21 and L2(81):41 are field automorphisms; this is for consistency with the GAP tables. 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. See here for details.

Version 2.0 created on 24th August 2000.
Last updated 23.02.04 by JNB.
Information checked to Level 0 on 24.08.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.