# ATLAS: Unitary group U4(2), Symplectic group S4(3)

Order = 25920 = 26.34.5.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of U4(2) = S4(3) are a and b, where a in class 2A and b has order 5 and ab has order 9.
Standard generators of the double cover 2.U4(2) = Sp4(3) are preimages A and B where B has order 5 and AB has order 9.

Standard generators of U4(2):2 = S4(3):2 are c and d, where c in class 2C and d has order 9 and cd has order 10.
Standard generators of either of the double covers 2.U4(2):2 are preimages C and D where D has order 9.

### Automorphisms

An outer automorphism of U4(2) of order 2 may be obtained by mapping (a, b) to (a, bbbb).

### Presentations

Presentations of U4(2) and U4(2):2 in terms of their standard generators are given below.

< a, b | a2 = b5 = (ab)9 = [a, b]3 = [a, bab]2 = 1 >.

< c, d | c2 = d9 = (cd2)8 = [c, d2]2 = [c, d3cd3] = 1 >.

### Representations

The representations of U4(2) available are:
• All primitive permutation permutation representations.
• a and b as permutations on 27 points.
• a and b as permutations on 36 points.
• a and b as permutations on 40 points - the cosets of N(3AB).
• a and b as permutations on 40 points - the cosets of 3^3:S4.
• a and b as permutations on 45 points.
• a and b as 4 × 4 matrices over GF(4) - the natural representation as U4(2).
• All faithful irreducibles in characteristic 3.
• a and b as 5 × 5 matrices over GF(3) - the natural representation as O5(3).
• a and b as 10 × 10 matrices over GF(3).
• a and b as 14 × 14 matrices over GF(3).
• a and b as 25 × 25 matrices over GF(3).
• a and b as 81 × 81 matrices over GF(3).
• a and b as 5 × 5 matrices over GF(25).
• a and b as 6 × 6 matrices over GF(5).
• a and b as 10 × 10 matrices over GF(25).
The representations of 2.U4(2) available are:
• A and B as permutations on 80 points.
• A and B as permutations on 240 points.
• All faithful irreducibles in characteristic 3.
• A and B as 4 × 4 matrices over GF(3) - the natural representation as Sp4(3).
• A and B as 16 × 16 matrices over GF(3).
• A and B as 40 × 40 matrices over GF(3).
The representations of U4(2):2 available are:
NB [24/11/99]: We have not yet rigorously checked that all representations of U4(2):2 and 2.U4(2).2 are in standard generators, but we believe this to be the case. [Quite a few of these representations were here prior to this date.]
• All faithful irreducibles in characteristic 2.
• c and d as 8 × 8 matrices over GF(2).
• c and d as 6 × 6 matrices over GF(2) - the representation as O6-(2).
• c and d as 14 × 14 matrices over GF(2).
• c and d as 40 × 40 matrices over GF(2).
• c and d as 64 × 64 matrices over GF(2) - the Steinberg representation.
• c and d as 5 × 5 matrices over GF(3) - the representation as O5(3).
• c and d as permutations on 36 points.
The representations of 2.U4(2):2 (ATLAS­version) available are:
• C and D as permutations on 240 points.
• C and D as 4 × 4 matrices over GF(3).

### Maximal subgroups

The maximal subgroups of U4(2) are as follows.
• 24:A5.
• S6 = A6:2.
• 31+2:2A4.
• 33:S4.
• 2.(A4 × A4).2.
The maximal subgroups of U4(2):2 are as follows.
• U4(2).
• 24:S5.
• S6 × 2.
• 31+2:2S4.
• 33:(S4 × 2).
• 2.(A4 × A4).2.2.

Last updated 25.05.00,
R.A.Wilson, R.A.Parker and J.N.Bray.