ATLAS: Symplectic group S_{10}(2)
Order = 24815256521932800 = 2^{25}.3^{6}.5^{2}.7.11.17.31.
Mult = 1.
Out = 1.
The following information is available for S_{10}(2):
Standard generators of S_{10}(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 S_{10}(2).
Finding generators
To find standard generators for S_{10}(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 S_{10}(2).
This algorithm is available in computer readable format:
finder for S_{10}(2).
Checking generators
To check that elements x and y of S_{10}(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(xz^{17}) = 3
This algorithm is available in computer readable format:
checker for S_{10}(2).
A presentation for S_{10}(2) in terms of its standard generators is given below.
< a, b  a^{2} = b^{11} = (ab)^{15} = (ab^{5})^{18} = [a, b]^{3} = [a, b^{2}]^{2} = [a, b^{3}]^{2} = [a, b^{4}]^{2} = [a, b^{5}]^{3} = [a, (ab)^{5}]^{2} = 1 >.
This presentation is available in Magma format as follows:
S10(2) on a and b.
The representations of S_{10}(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.

Dimension 155 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 155 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 155 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 155 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 155 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 155 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 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).
The maximal subgroups of S_{10}(2) include the following.

O10(2):2.
Order: 50030759116800.
Index: 496.

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

2^{9}:S_{8}(2).
Order: 24257337753600.
Index: 1023.

2^{15}:L_{5}(2).
Order: 327659028480.
Index: 75735.

[2^{15}]:(S_{6}(2) × S_{3}).
Order: 285380444160.
Index: 86955.

S_{8}(2) × S_{3}.
Order: 284265676800.
Index: 87296.

[2^{18}]:(S_{6} × L_{3}(2)).
Order: 31708938240.
Index: 782595.

[2^{18}]:(S_{3} × A_{8}).
Order: 31708938240.
Index: 782595.

S_{6}(2) × S_{6}.
Order: 1045094400.
Index: 23744512.

L_{2}(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.