ATLAS: Unitary group U_{3}(4)
Order = 62400 = 2^{6}.3.5^{2}.13.
Mult = 1.
Out = 4.
The following information is available for U_{3}(4):
Standard generators of U_{3}(4) are a and b where
a has order 2, b has order 3 and ab has order 13.
Standard generators of U_{3}(4):2 are
c
and d where
c has order 2,
d has order 3,
cd has order 8,
cdcdd has order 13
and cdcdcdcddcdcddcdd has order 10.
NB: Of course, c is in class 2B.
Standard generators of U_{3}(4):4 are e and f where
e is in class 2A, f is in class 4B or 4B', ef has order 12
and efefffeff has order 6.
NB: These conditions distinguish between classes 4B and 4B'. Classes 4B and
4B' are the classes for which the ATLAS class 4B is proxy.
With these conditions, f is conjugate to the *2 automorphism.
A generating outer automorphism of U_{3}(4) may be obtained by mapping
(a, b)
to ((ab)^{2}a(ab)^{2}, (ab^{1})^{5}b(ab^{1})^{5}).
An outer automorphism of U_{3}(4) of order 2 is given by mapping
(a, b) to (a, b^{1})
Presentations of U_{3}(4), U_{3}(4):2 and U_{3}(4):4 in terms of their standard generators are given below.
< a, b  a^{2} = b^{3} =
(ab)^{13} = [a, b]^{5} =
[a, babab]^{3} = 1 >.
< c, d  c^{2} = d^{3} =
(cd)^{8} = [c, d]^{13} =
[c, dcdcdcd^{1}cdcd]^{2} =
[c, d^{1}cdcd]^{5} = 1 >.
< e, f  e^{2} = f^{4} =
(ef)^{12} = [e, f]^{5} =
(ef^{2})^{10} =
efefef^{2}efef^{2}ef^{1}ef^{2}efefef^{1}ef^{2}ef^{1}ef^{2} = 1 >.
These presentations are available in Magma format as follows:
U3(4) on a and b,
U3(4):2 on c and d and
U3(4):4 on e and f.
The representations of U_{3}(4) available are:
 All primitive permutation representations.

Permutations on 65 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 208 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 416 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1600 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful absolute irreducibles in characteristic 2.

Dimension 3a over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 3b over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 3c over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 3d over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8a over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8b over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9a over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9b over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9c over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9d over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 24a over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 24b over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 24c over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 24d over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the Steinberg representation.
 All other faithful irreducibles in characteristic 2.

Dimension 6a over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 3 over GF(16).

Dimension 6b over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 3 over GF(16).

Dimension 12 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 3 over GF(16).

Dimension 16 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 8 over GF(4).

Dimension 18a over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 9 over GF(16).

Dimension 18b over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 9 over GF(16).

Dimension 36 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 9 over GF(16).

Dimension 48a over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 24 over GF(16).

Dimension 48b over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 24 over GF(16).

Dimension 96 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 24 over GF(16).
 Essentially all faithful irreducibles in characteristic 3.

Dimension 12 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 52e over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 13 over GF(81).

Dimension 64 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 75a over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 75b over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 75c over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 75d over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 39 over GF(9).

Dimension 208 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 52 over GF(81).
 Essentially all faithful irreducibles in characteristic 5.

Dimension 12 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 39 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 150a over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 75 over GF(625).

Dimension 150b over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 75 over GF(625).

Dimension 300 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 75 over GF(625).
 Essentially all faithful irreducibles in characteristic 13.

Dimension 12 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 52e over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 13 over GF(28561).

Dimension 63 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65a over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 39 over GF(169).

Dimension 208 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 52 over GF(28561).

Dimension 260 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 really dimension 65 over GF(28561).
 a and
b as
39 × 39 matrices over GF(169).
The representations of U_{3}(4):2 available are:
 All primitive permutation representations.

Permutations on 65 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 208 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 416 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 primitive.

Permutations on 1600 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 c and
d as
6 × 6 matrices over GF(4).
The representations of U_{3}(4):4 available are:
 All primitive permutation representations.

Permutations on 65 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Permutations on 208 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Permutations on 416 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
 on the cosets of the maximal subgroup 5^2:(4 × S3).

Permutations on 1600 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
 All faithful irreducibles in characteristic 2.

Dimension 12 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 16 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 36 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 64 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 96 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
 All faithful irreducibles in characteristic 3 with character in ABC.

Dimension 12 over GF(9):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 24 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 52 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 64 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 78 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 208 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 300 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
 All faithful irreducibles in characteristic 5 with character in ABC.

Dimension 12 over GF(5):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 39 over GF(5):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 65 over GF(5):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 300 over GF(5):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
 All faithful irreducibles in characteristic 13 with character in ABC.

Dimension 12 over GF(13):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 52 over GF(13):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 63 over GF(13):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 65 over GF(13):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 78 over GF(13):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 208 over GF(13):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 260 over GF(13):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
The maximal subgroups of U_{3}(4) are as follows.
The maximal subgroups of U_{3}(4):2 are as follows.

U_{3}(4), with standard generators
(cd)^{4}, d.

2^{2+4}:(3 × D_{10}), with generators
c, cdcd^{2}cdcd^{2}cdcd.

D_{10} × A_{5}, with generators
c, (cdcd^{2}cd^{2}cd)^{2}.

5^{2}:D_{12}, with generators
c, dcdcd^{2}cdcd^{2}cd^{2}cdcdcdcd^{2}.

13:6 = F_{78}, with generators
c, dcdcd^{2}cdcdcd.
The maximal subgroups of U_{3}(4):4 are as follows.

U_{3}(4):2, with standard generators
(eff)^{5}, ((ef)^{4})^{fefffef}.

2^{2+4}:(3 × D_{10}).2, with generators
efef^{2}ef^{1}efe, f.

(D_{10} × A_{5}).2, with generators
(ef)^{6}, f.

5^{2}:(4 × S_{3}), with generators
e, (f^{2}ef^{1})^{3}.

13:12 = F_{156}, with generators
ef^{2}ef^{2}ef^{2}efef^{1}efef^{2}e, f.
NB: Maps between the various extensions of U3(4) have not been checked for compatibility with the class definitions (or even compatibility with each other).
Some conjugacy classes U_{3}(4) are as follows.
 1A: identity.
 2A: a.
 3A: b.
 13A: ab.
Some conjugacy classes U_{3}(4):2 are as follows.
 1A: identity.
 13AB: cdcdd.  compatible with U34d2G1max1W1 and ab being in class 13A.
 8A: cd.
 8B: cdd.
Some conjugacy classes U_{3}(4):4 are as follows.
 1A: identity.^{ }
 2A: e.
 4B: f.
 12A: ef.
 16A: effefff.
 16B: efeffeff.
 4B': fff.
 12A': efff.
 16A': efeff.
 16B': effeffefff.
Choices made:
 ab is in class 13A.
 cdcdd is in class 13AB.
 f is in class 4B (rather than 4B').
 Everything else should follow from definitions in the ABC (with consistency between the various extensions).
Go to main ATLAS (version 2.0) page.
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See here for details.
Version 2.0 created on 27th August 2004, from a version 1 file last updated on 12th April 2000.
Last updated 15.04.05 by RAW.
R.A.Wilson, S.J.Nickerson and J.N.Bray.