ATLAS: Unitary group U_{5}(2)
Order = 13685760 = 2^{10}.3^{5}.5.11.
Mult = 1.
Out = 2.
The following information is available for U_{5}(2):
Standard generators of U_{5}(2) are a and b where
a is in class 2A, b has order 5 and ab has order 11.
(Wlog we take ab in 11A.)
Standard generators of U_{5}(2).2 are
c
and d where
c has order 2 (so is in class 2C),
d has order 4 (so is in class 4D),
cd has order 11
and cdcdd has order 4.
(Wlog we take cdcdcddcdcdcdcddcdcdd in class 16B.)
An outer automorphism may be obtained by mapping
(a, b) to (a, (abb)^5b(abb)^5).
Another outer automorphism may be obtained by mapping
(a, b) to (a, bbbb).
Both these automorphisms have order 2.
We may obtain standard generators of U_{5}(2):2 as
c = u and
d = u(ab^{2})^{2}(ab^{3})^{3}(ab^{2})^{2}, where u is the automorphism
u : (a, b) >
(a, b^{1}).
Presentations of U_{5}(2) and U_{5}(2):2 on their
standard generators are given below.
< a, b  a^{2} = b^{5} =
(ab)^{11} = [a, b]^{3} =
[a, b^{2}]^{3} =
[a, bab]^{3} =
[a, bab^{2}]^{3} = 1 >.
< c, d  c^{2} = d^{4} =
(cd)^{11} = (cd^{2})^{6} =
(cdcd^{2})^{4} = [c, d]^{9} =
(cd^{1}(cdcdcd^{1})^{2}cd^{2})^{2} =
[d^{2}, cdcdc]^{3} = 1 >.
These presentations are available in Magma format as follows:
U_{5}(2) on a and b and
U_{5}(2):2 on c and d.
The representations of U_{5}(2) available are:
 All primitive permutation representations.

Permutations on 165 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 176 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 297 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1408 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 3520 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 20736 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 2.

Dimension 5 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 5 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 24 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 74 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 3.

Dimension 10 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 44 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 100 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 110 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 5.

Dimension 43 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 176 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 a and
b as
11 x 11 matrices over GF(25).
 Some irreducibles in characteristic 11.

Dimension 44 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 119 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 176 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 a and
b as
11 x 11 matrices over GF(121).
 Some irreducibles in characteristic 7 (not dividing the group order!).
 a and
b as
10 x 10 matrices over GF(7).
 a and
b as
11 x 11 matrices over GF(7).
 Some irreducibles in characteristic 0.

Dimension 10 over Z[i]:
a and b (Magma).

Dimension 10 over Z[w]:
a and b (Magma).

Dimension 10 over Z[i2]:
a and b (Magma).

Dimension 11a over Z[w]:
a and b (Magma).

Dimension 11b over Z[w]:
a and b (Magma).

Dimension 20 over Z  reducible over C:
a and b (Magma).
The representations of U_{5}(2):2 available are:
 All faithful primitive permutation representations.

Permutations on 165 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 176 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 297 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 1408 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 3520 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 20736 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 2.

Dimension 10 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

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

Dimension 24 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 80 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 80 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 74 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 320 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 560 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 1024 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 10 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some irreducibles in characteristic 0.

Dimension 22 over Z:
c and d (Magma).
The maximal subgroups of U_{5}(2) are as follows.

2^{1+6}.3^{1+2}.2A_{4}, with generators
abab^{1}ab, ab^{2}ab^{2}ab^{2}
centralising a.

3 × U_{4}(2), with generators
here, mapping to standard generators of U_{4}(2).
A conjugate of this group has generators
ab^{2}, (ab^{3})^{5}
centralising (ab^{2})^{5}.

2^{4+4}:(3 × A_{5}), with generators
ab^{2}ab^{3}abababab^{4}a, b.

3^{4}:S_{5}, with generators
a, b^{ab}.

S_{3} × 3^{1+2}:2A_{4}.

L_{2}(11), with standard generators
here.
The maximal subgroups of U_{5}(2):2 are as follows.

U_{5}(2), with standard generators
(dcdcd^{2}cd)^{6}, (cd^{2}cd^{3}cdcdcdc)^{3}, and generators
d^{2}, cd^{1}.

2^{1+6}.3^{1+2}.2S_{4}, with generators
c, dcd^{1}cd^{1}cdcdcd^{2}.

(3 × U_{4}(2)):2, with generators
c, dcd^{2}cd^{2}.

2^{4+4}:(3 × A_{5}):2, with generators
cdcdcd^{1}c, d.

3^{4}:(S_{5} × 2), with generators
c, dcdcd^{1}cd^{1}cdcd.

S_{3} × 3^{1+2}:2S_{4}.

L_{2}(11):2, with generators
d^{2}, d^{cdcdc}.
Generators of the maximal cyclic subgroups of U_{5}(2) are
available here, and a program to obtain
representatives of all conjugacy classes of U_{5}(2) from these
is available here.
Generators of the maximal cyclic subgroups of U_{5}(2):2 are
available here, and a program to obtain
representatives of all conjugacy classes of U_{5}(2):2 from these
is available here.
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old U5(2) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 14th December 2001.
Last updated 10.11.03 by JNB.
Information checked to
Level 0 on 14.12.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.