# 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 10-dimensional representation of 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 >.

### Representations

The representations of S10(2) available are:
• a and b as permutations on 496 points.
• a and b as permutations on 528 points.
• a and b as permutations on 1023 points.
• a and b as permutations on 992 points - imprimitive.
• a and b as permutations on 1056 points - imprimitive.
• a and b as 10 × 10 matrices over GF(2) - the natural representation.
• a and b as 32 × 32 matrices over GF(2).
• a and b as 44 × 44 matrices over GF(2).
• a and b as 100 × 100 matrices over GF(2).

### 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.

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

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

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

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

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

• L2(32):5.
Order: 163680.
Index: 151608360960.