# ATLAS: Symplectic group S10(2)

Order = 24815256521932800 = 225.36.52.7.11.17.31.
Mult = 1.
Out = 1.

The following information is available for S10(2):

### Standard generators

Standard generators of S10(2) are a and b where a is in class 2A, b has order 11 and ab has order 15.
NB: Class 2A is the class of transvections in the natural 10­dimensional representation of S10(2).

### Black box algorithms

#### Finding generators

To find standard generators for S10(2):
• Find an element of order 34. Its 17th power, x say, is in class 2A.
[The probability of success at each attempt is 1 in 17.]
• Find an element of order 11 or 33. This powers up to y of order 11.
[The probability of success at each attempt is 1 in 11.]
• Find a conjugate a of x and a conjugate b of y such that ab has order 15.
[The probability of success at each attempt is 1 in 31.]
• Now a and b are standard generators of S10(2).
This algorithm is available in computer readable format: finder for S10(2).

#### Checking generators

To check that elements x and y of S10(2) are standard generators:

• Check o(x) = 2
• Check o(y) = 11
• Check o(xy) = 15
• Let z = xyxyy(xy)3(xyy)4
• Check o(z) = 34
• Check o(xz17) = 3
This algorithm is available in computer readable format: checker for S10(2).

### Presentation

A presentation for S10(2) in terms of its standard generators is given below.

< a, b | a2 = b11 = (ab)15 = (ab5)18 = [a, b]3 = [a, b2]2 = [a, b3]2 = [a, b4]2 = [a, b5]3 = [a, (ab)5]2 = 1 >.

This presentation is available in Magma format as follows: S10(2) on a and b.

### Representations

The representations of S10(2) available are:
• Some faithful permutation representations.
• Permutations on 496 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).
• Permutations on 992 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1023 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1056 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 2.
• Dimension 10 over GF(2) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 32 over GF(2): 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).
• Some faithful irreducibles in characteristic other than 2.
• Some faithful irreducibles in characteristic 0
• Dimension 155 over Z: a and b (GAP).
• Dimension 187 over Z: a and b (GAP).
• Dimension 341 over Z: a and b (GAP).

### Maximal subgroups

The maximal subgroups of S10(2) include the following.
• O10-(2):2.
Order: 50030759116800.
Index: 496.

• O10+(2):2.
Order: 46998591897600.
Index: 528.

• 29:S8(2).
Order: 24257337753600.
Index: 1023.

• 215:L5(2).
Order: 327659028480.
Index: 75735.

• :(S6(2) × S3).
Order: 285380444160.
Index: 86955.

• S8(2) × S3.
Order: 284265676800.
Index: 87296.

• :(S6 × L3(2)).
Order: 31708938240.
Index: 782595.

• :(S3 × A8).
Order: 31708938240.
Index: 782595.

• S6(2) × S6.
Order: 1045094400.
Index: 23744512.

• L2(32):5.
Order: 163680.
Index: 151608360960. Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old S10(2) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 22nd May 1999.
Last updated 13.01.05 by SJN.
Information checked to Level 1 on 22.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.