# ATLAS: Orthogonal group O10+(2)

Order = 23499295948800 = 220.35.52.7.17.31.
Mult = 1.
Out = 2.

The following information is available for O10+(2):

### Standard generators

Standard generators of O10+(2) are a and b where a is in class 2A, b is in class 20A and ab has order 21.
Standard generators of O10+(2):2 are c and d where c is in class 2E, d has order 16 and cd has order 45.

### Automorphisms

An outer automorphism of O10+(2) can be taken to map (a, b) to (a, b-1).

### Black box algorithms

To find standard generators for O10+(2):
• Find any element of order 60. It powers up to x in class 2A and y in class 20A.
[The probability of success at each attempt is 1 in 30.]
• Find a conjugate a of x and a conjugate b of y such that ab has order 21.
[The probability of success at each attempt is 32 in 1581 (about 1 in 49).]
• Now a and b are standard generators for O10+(2).
To find standard generators for O10+(2).2:
• Find any element of order 34. It powers up to x in class 2E.
[The probability of success at each attempt is 1 in 17 (or 2 in 17 if you can restrict your search to outer elements only).]
• Find any element, y say, of order 16.
[The probability of success at each attempt is 1 in 32 (or 1 in 16 if you can restrict your search to outer elements only).]
• Find a conjugate c of x and a conjugate d of y such that cd has order 45.
[The probability of success at each attempt is 2 in 31 (about 1 in 16).]
• Now c and d are standard generators for O10+(2):2.

### Presentations

Presentations of O10+(2) and O10+(2):2 on their standard generators are given below:

< a, b | a2 = b20 = (ab)21 = (ab2)17 = . . . = 1 >.

< c, d | c2 = d16 = (cd)45 = [c, d]3 = [c, d2]2 = [c, d3]3 = [c, d4]2 = [c, d5]2 = [c, d6]2 = [c, d7]2 = (cd8)4 = (cd2cd2cd-1)9 = 1 >.

The relations (cd)45 = [c, d]3 = 1 in the O10+(2):2 presentation are redundant.
These presentations are available in Magma format as follows: O10+(2):2 on c and d.

### Representations

The representations of O10+(2) available are:
• Primitive permutation representations.
• Permutations on 496 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 527 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 2295a points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 2295b points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 19840 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 23715 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 39680 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive.
• Faithful irreducibles in characteristic 2.
• Dimension 10 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the natural representation.
• Dimension 16a over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - a ½-spin representation.
• Dimension 16b over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the other ½-spin representation.
• 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 144a over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - in 10 × 16a.
• Dimension 144b over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - in 10 × 16b.
• Dimension 164 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 320 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 416a over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - in 44 × 16a.
• Dimension 416b over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - in 44 × 16b.
• Dimension 670 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Faithful irreducibles in characteristic 3.
The representations of O10+(2):2 available are:
• Primitive permutation representations.
• Permutations on 496 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 527 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 4590 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - imprimitive.
• Faithful irreducibles in characteristic 2.
• Dimension 10 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - the natural representation.
• Dimension 32 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - the spin representation.
• 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 288 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 320 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).
• Dimension 832 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.
The maximal subgroups of O10+(2):2 are as follows.

Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old O10+(2) page - ATLAS version 1.
Anonymous ftp access is also available. See here for details.

Version 2.0 created on 23rd January 2004.
Last updated 26.01.04 by JNB/SJN.
Information checked to Level 0 on 13.10.03 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.