ATLAS: Unitary group U_{3}(4)
Order = 62400 = 2^{6}.3.5^{2}.13.
Mult = 1.
Out = 4.
Standard generators
Standard generators of U_{3}(4) are a and b where
a has order 2, b has order 3 and ab has order 13.
A generating outer automorphism may be obtained by
mapping
(a, b)
to ((ab)^-2a(ab)^2, (abb)^-5b(abb)^5).
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.
Presentations
< 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 >.
Representations
The representations of U3(4) available are
- a and
b as
3 × 3 matrices over GF(16) - the natural representation.
- a and
b as
6 × 6 matrices over GF(4) - obtained from the above by regarding GF(16) as a 2-space
over GF(4).
- a and
b as
8 × 8 matrices over GF(4) - this one is absolutely irreducible.
- a and
b as
16 × 16 matrices over GF(2) - the Steinberg representation.
- a and
b as
12 × 12 matrices over GF(3).
- a and
b as
64 × 64 matrices over GF(3).
- a and
b as
12 × 12 matrices over GF(5).
- a and
b as
39 × 39 matrices over GF(5).
- a and
b as
65 × 65 matrices over GF(5).
- a and
b as
39 × 39 matrices over GF(169).
- a and
b as
63 × 63 matrices over GF(13).
- a and
b as
65 × 65 matrices over GF(13).
- a and
b as
208 x 208 matrices over GF(13) - really dimension 52 over GF(28561).
- a and
b as
permutations on 65 points.
- a and
b as
permutations on 208 points.
- a and
b as
permutations on 416 points.
The representations of U3(4):2 available are
- c and
d as
6 × 6 matrices over GF(4).
The representations of U3(4):4 available are
NB: Changed to standard generators on 26.05.98.
- e and
f as
12 × 12 matrices over GF(2).
- e and
f as
12 × 12 matrices over GF(5).
- e and
f as
permutations on 65 points.
- e and
f as
permutations on 208 points.
- e and
f as
permutations on 416 points, on the cosets of the maximal subgroup
5^2:(4 × S3).
- e and
f as
permutations on 1600 points.
Maximal subgroups
The maximal subgroups of U3(4) are as follows.
- 2^2+4:15
- 5 × A5
- 5^2:S3
- 13:3
The maximal subgroups of U3(4):2 are as follows.
- U3(4)
- 2^2+4:(3 × D10)
- D10 × A5
- 5^2:D12
- 13:6
The maximal subgroups of U3(4):4 are as follows.
- U3(4):2
- 2^2+4:(3 × D10).2
- (D10 × A5).2
- 5^2:(4 × S3)
- 13:12
Return to main ATLAS page.
Last updated 12.04.00
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk