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
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.
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Last updated 02.09.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk