ATLAS: Orthogonal group O₁₀^-(2)

Order = 25015379558400 = 2²⁰.3⁶.5².7.11.17.
Mult = 1.
Out = 2.

The following information is available for O₁₀^-(2):

Standard generators.
Black box algorithms.
Representations.
Maximal subgroups.
Conjugacy classes.
Checks applied.

Standard generators

Standard generators of O₁₀^-(2) are a and b, where a is in class 2A, d is in class 5B, ab has order 33 and abababb has order 18. The last condition is equivalent to: ababbbb has order 3.

Standard generators of O₁₀^-(2):2 are c and d, where c is in class 2E, d has order 11 and cd has order 36.
N.B. The old generators for 010-(2):2 that were here before 20.11.97 are no longer available.

Representations

The representations of O₁₀^-(2) available are:

Permutations on 495 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 528 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 10 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 16 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 16 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 44 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 100 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 164 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 670 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

The representations of O₁₀^-(2):2 available are:

Permutations on 495 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Permutations on 528 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Permutations on 104448 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP), c and d (Magma).
Dimension 10 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 32 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 44 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 100 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 164 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 670 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

Maximal subgroups

The maximal subgroups of O₁₀⁻(2) are as follows.

2⁸:O₈^-(2).
S₈(2).
2¹⁺¹²:(U₄(2) × S₃).
(3 × O₈⁺(2)):2.
2⁶⁺⁸:(3 × A₈).
2³⁺¹²:(L₃(2) × A₅).
A₁₂.
3 × U₅(2).
(A₅ × A₈):2.
(S₃ × S₃ × U₄(2)):2.

The maximal subgroups of O₁₀⁻(2):2 are as follows.

O₁₀^-(2).
2⁸:O₈^-(2):2.
S₈(2) × 2.
2¹⁺¹²:(U₄(2):2 × S₃).
S₃ × O₈⁺(2):2.
2⁶⁺⁸:(S₃ × A₈).
2³⁺¹²:(L₃(2) × S₅).
S₁₂.
M₁₂:2.
(3 × U₅(2)):2.
S₅ × S₈.
(S₃ × S₃):2 × U₄(2):2.

Conjugacy classes

A set of generators for the maximal cyclic subgroups of O₁₀^-(2) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. These classes are compatible with the Atlas of Brauer Characters.

A set of generators for the maximal cyclic subgroups of O₁₀^-(2):2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. These classes are compatible with the Atlas of Brauer Characters.

Checks applied

Check	Date	By whom
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	26.01.01	RAW
All representations are in standard generators

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