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.
Presentations.
Representations.
Maximal subgroups.

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:

Dimension 16 over GF(3): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).

The representations of O₈^-(3).2₃ available are:

Permutations on 1066 points: g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
Permutations on 1066 points: g₂ and h₂ (Meataxe), g₂ and h₂ (Meataxe binary), g₂ and h₂ (GAP).
Dimension 8 over GF(3): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).

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:

3⁶:2.U₄(3):2₁.
O₇(3):2.
O₇(3):2.
3³⁺⁶:(L₃(3) × 4).
(4 × L₄(3)):2₂.
L₂(81):2₁.

The maximal subgroups of O₈^-(3):2₁ are as follows - INCOMPLETE:

O₈^-(3).
3⁶:2.U₄(3):[?](2²)₁₂₂.
O₇(3):2 × 2.
O₇(3):2 × 2.
3³⁺⁶:(L₃(3) × D₈).
D₈ × L₄(3):2₂.
L₂(81):4₁.

The maximal subgroups of O₈^-(3):2₂ are as follows - INCOMPLETE:

O₈^-(3).
3⁶:2.U₄(3):4.
3³⁺⁶:(L₃(3) × 8).
L₄(3).D₁₆.
L₂(81):2₁ × 2.

The maximal subgroups of O₈^-(3).2₃ are as follows - INCOMPLETE:

O₈^-(3).
3⁶:2.U₄(3).(2²)₁₃₃.
3³⁺⁶:(L₃(3) × Q₈).
L₄(3).SD₁₆.
L₂(81):4₁.

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).[2⁵].
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.

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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.

ATLAS: Orthogonal group O8-(3)

Standard generators

Representations

Maximal subgroups

ATLAS: Orthogonal group O₈^-(3)