ATLAS: Unitary group U_{3}(5)
Order = 126000 = 2^{4}.3^{2}.5^{3}.7.
Mult = 3.
Out = S_{3}.
The following information is available for U_{3}(5):
Standard generators of U_{3}(5) are a and b where
a has order 3, b is in class 5A and ab has order 7.
Standard generators of 3.U_{3}(5) are preimages A
and B where B has order 5 and AB has order 7.
Standard generators of U_{3}(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.U_{3}(5):2 are preimages C
and D where D has order 4.
Standard generators of U_{3}(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.U_{3}(5).3 are preimages E and
F where E has order 2.
Standard generators of U_{3}(5):S_{3} 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.U_{3}(5).S_{3} are preimages
G and H. (All such preimages are automorphic.)
Presentations of U_{3}(5), U_{3}(5):2, U_{3}(5):3 and
U_{3}(5):S_{3} on their standard generators are given below.
< a, b  a^{3} = b^{5} =
(ab)^{7} = (ab^{1})^{7} =
aba^{1}b^{2}aba^{1}bab^{2}a^{1}b = 1 >.
< c, d  c^{2} = d^{4} =
(cd)^{10} =
(cdcd^{1}cd^{2})^{2} =
[c, dcd]^{4} =
(cdcdcdcd^{2})^{7} = 1 >.
< e, f  e^{2} = f^{3} =
(ef)^{15} = [e, f]^{6} =
[e, (ef)^{4}(ef^{1})^{3}(ef)^{4}] =
[e, (ef)^{3}(ef^{1})^{6}(ef)^{3}] = 1 >.
< g, h  g^{2} = h^{3} =
(gh)^{8} = [g, h]^{12} =
(ghghgh^{1})^{6} =
[g, hgh]^{6} =
(ghghgh^{1}ghgh^{1})^{10} = 1 >.
These presentations are available in Magma format as follows:
U_{3}(5):2 on c and d.
The representations of U_{3}(5) available are:
 Permutation representations, including all primitive ones.

Permutations on xxx points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 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 Z (χ_{4} 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.U_{3}(5) available are:
 A and
B as
3 x 3 matrices over GF(25)  the natural representation.
The representations of U_{3}(5):2 available are:
 Permutation representations, including all faithful primitive ones.

Permutations on 50 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 126 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 175 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 525 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 750 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 2.

Dimension 20 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 28 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 56 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 104 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 288 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 3, up to tensoring with linear irreducibles.

Dimension 20 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 21 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 28 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 56 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 84 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 126 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 252 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 288 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 5, up to tensoring with linear irreducibles.

Dimension 8 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 19 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 20 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 63 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 70 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 125 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 7, up to tensoring with linear irreducibles.

Dimension 20 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 21 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 28 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 56 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 84 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 105 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 124 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 126 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 252 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 3.U_{3}(5):2 available are:
 C and
D as
6 x 6 matrices over GF(5).
The maximal subgroups of U_{3}(5) are as follows.
The maximal subgroups of U_{3}(5):2 are as follows.

U_{3}(5), with standard generators
(ababbab)^4, (abababbababb)^4.

S_{7}, with generators
c, d^{1}cdcdcd,
and standard generators
c, d(cd^{2}cd^{3}cd)^{2}cd.

5^{1+2}:8:2, with generators
c, cdcdcd^{2}cd^{2}._{ }

A_{6}.2^{2}, with standard
generators c, d^{2}cdcd^{2}.

2.S_{5}.2, with generators
c, (cdcd^{2}cd)^{2}.

L_{2}(7):2, with standard generators
c, (dcd)^{4}.
The maximal subgroups of U_{3}(5):3 are as follows.
 U_{3}(5)
 5^{1+2}:24
 2.S_{5} × 3
 6^{2}:S_{3}
 3^{2}:2A_{4}
 7:3 × 3
The maximal subgroups of U_{3}(5):S_{3} are as follows.
 U_{3}(5):3
 U_{3}(5):2
 5^{1+2}:24:2
 (3 × 2.S_{5}).2
 6^{2}:D_{12}
 3^{2}:2S_{4}
 (7:3 × 3):2
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.
cdcdcd^{3} 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.