ATLAS: Unitary group U3(5)

Order = 126000 = 24.32.53.7.
Mult = 3.
Out = S3.

The following information is available for U3(5):

Standard generators

Standard generators of U3(5) are a and b where a has order 3, b is in class 5A and ab has order 7.
Standard generators of 3.U3(5) are preimages A and B where B has order 5 and AB has order 7.

Standard generators of U3(5):2 are c and d where c is in class 2B, d is in class 4A, cd has order 10 and cdcdddcdd has order 2.
Standard generators of 3.U3(5):2 are preimages C and D where D has order 4.

Standard generators of U3(5):3 are e and f where e is in class 2A, f is in class 3C/C' and ef has order 15.
Standard generators of any 3.U3(5).3 are preimages E and F where E has order 2.

Standard generators of U3(5):S3 are g and h where g is in class 2B, h is in class 3C, gh has order 8 and ghghh has order 12.
Standard generators of any 3.U3(5).S3 are preimages G and H. (All such preimages are automorphic.)

Presentations

Presentations of U3(5), U3(5):2, U3(5):3 and U3(5):S3 on their standard generators are given below.

< a, b | a3 = b5 = (ab)7 = (ab-1)7 = aba-1b2aba-1bab2a-1b = 1 >.

< c, d | c2 = d4 = (cd)10 = (cdcd-1cd2)2 = [c, dcd]4 = (cdcdcdcd2)7 = 1 >.

< e, f | e2 = f3 = (ef)15 = [e, f]6 = [e, (ef)4(ef-1)3(ef)4] = [e, (ef)3(ef-1)6(ef)3] = 1 >.

< g, h | g2 = h3 = (gh)8 = [g, h]12 = (ghghgh-1)6 = [g, hgh]6 = (ghghgh-1ghgh-1)10 = 1 >.

These presentations are available in Magma format as follows: U3(5):2 on c and d.

Representations

The representations of U3(5) available are:
• Permutation representations, including all primitive ones.
• a and b as permutations on 50 points.
• All faithful irreducibles in characteristic 2.
• a and b as 20 x 20 matrices over GF(2).
• a and b as 28 x 28 matrices over GF(2).
• a and b as 28 x 28 matrices over GF(2).
• a and b as 28 x 28 matrices over GF(2).
• a and b as 104 x 104 matrices over GF(2).
• a and b as 144 x 144 matrices over GF(2).
• a and b as 144 x 144 matrices over GF(2).
• All faithful irreducibles in characteristic 3.
• a and b as 20 x 20 matrices over GF(3).
• a and b as 21 x 21 matrices over GF(3).
• a and b as 28 x 28 matrices over GF(3).
• a and b as 28 x 28 matrices over GF(3).
• a and b as 28 x 28 matrices over GF(3).
• a and b as 84 x 84 matrices over GF(3).
• a and b as 126 x 126 matrices over GF(3).
• a and b as 126 x 126 matrices over GF(3).
• a and b as 126 x 126 matrices over GF(3).
• a and b as 144 x 144 matrices over GF(9).
• a and b as 144 x 144 matrices over GF(9).
• All faithful irreducibles in characteristic 5.
• a and b as 8 x 8 matrices over GF(5).
• a and b as 10 x 10 matrices over GF(25).
• a and b as 10 x 10 matrices over GF(25).
• a and b as 19 x 19 matrices over GF(5).
• a and b as 35 x 35 matrices over GF(25).
• a and b as 35 x 35 matrices over GF(25).
• a and b as 63 x 63 matrices over GF(5).
• a and b as 125 x 125 matrices over GF(5) - the Steinberg representation.
• All faithful irreducibles in characteristic 7.
• a and b as 20 x 20 matrices over GF(7).
• a and b as 21 x 21 matrices over GF(7).
• a and b as 28 x 28 matrices over GF(7).
• a and b as 28 x 28 matrices over GF(7).
• a and b as 28 x 28 matrices over GF(7).
• a and b as 84 x 84 matrices over GF(7).
• a and b as 105 x 105 matrices over GF(7).
• a and b as 124 x 124 matrices over GF(7).
• a and b as 126 x 126 matrices over GF(7).
• a and b as 126 x 126 matrices over GF(49).
• a and b as 126 x 126 matrices over GF(49).
• Some faithful irreducibles in characteristic 0
• Dimension 21 over Z: a and b (GAP).
• Dimension 28 over Z4 in the ATLAS): a and b (GAP).
• Dimension 105 over Z: a and b (GAP).
• Dimension 125 over Z: a and b (GAP).
The representations of 3.U3(5) available are:
• A and B as 3 x 3 matrices over GF(25) - the natural representation.
The representations of U3(5):2 available are:
• Permutation representations, including all faithful primitive ones.
• All faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3, up to tensoring with linear irreducibles.
• All faithful irreducibles in characteristic 5, up to tensoring with linear irreducibles.
• All faithful irreducibles in characteristic 7, up to tensoring with linear irreducibles.
The representations of 3.U3(5):2 available are:
• C and D as 6 x 6 matrices over GF(5).

Maximal subgroups

The maximal subgroups of U3(5) are as follows.
The maximal subgroups of U3(5):2 are as follows.
The maximal subgroups of U3(5):3 are as follows.
• U3(5)
• 51+2:24
• 2.S5 × 3
• 62:S3
• 32:2A4
• 7:3 × 3
The maximal subgroups of U3(5):S3 are as follows.
• U3(5):3
• U3(5):2
• 51+2:24:2
• (3 × 2.S5).2
• 62:D12
• 32:2S4
• (7:3 × 3):2

Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy have been dealt with.
We have made a choice of the ordering of classes 5B/C/D, which may change if we ever decide on a consistent convention for such things.

cdcdcd3 is taken to lie in class 20A. Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old U3(5) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 27th October 2003, from Version 1 file last modified on 12.04.00.
Last updated 03.03.04 by SJN.
Information checked to Level 1 on 27.10.03 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.