# ATLAS: Symplectic group S4(5)

Order = 4680000.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of S4(5) are a and b where a is in class 2B, b is in class 3B, and ab is in class 13C. The last condition is equivalent to: ab has order 13 and ababb has order 12.
Standard generators of S4(5).2 are c and d where c is in class 2D, b is in class 3B, and ab is in class 26A. The last condition is equivalent to: cd has order 13 and cdcdd has order 10.

### Representations

The representations of S4(5) available are
• a and b as 5 x 5 matrices over GF(5) - the representation as O5(5).
• a and b as 12 x 12 matrices over GF(4).
• a and b as 13 x 13 matrices over GF(9).
• a and b as permutations on 156 points, on the cosets of 5^1+2:4A5.
• a and b as permutations on 156 points, on the cosets of 5^3:A5.4
• a and b as permutations on 300 points.
• a and b as permutations on 325 points.
The representations of 2.S4(5) available are
• A and B as 4 x 4 matrices over GF(5) - the representation as Sp4(5).
• A and B as permutations on 624 points.
The representations of S4(5):2 available are
• c and d as 5 x 5 matrices over GF(5) - the representation as O5(5).
• c and d as 24 x 24 matrices over GF(2).
• c and d as 40 x 40 matrices over GF(2).
• c and d as 64 x 64 matrices over GF(2).
• c and d as 104 x 104 matrices over GF(2).
• c and d as 104 x 104 matrices over GF(2).
• c and d as 26 x 26 matrices over GF(3).
The representation of 2.S4(5):2 available is
• none

### Maximal subgroups

The maximal subgroups of S4(5) are as follows. Words in preparation - mostly wrong at the moment!
• 5^1+2:4A5, with generators here.
• 5^3:(2 x A5).2, with generators here.
• L2(25).2, with generators here.
• 2.(A5 x A5).2, with generators here.
• 2^4:A5, with generators here.
• S3 x S5, with generators here.
• (2 x 2 x A5):2, with generators here.
• A6, with generators here.