ATLAS: Unitary group U3(3), Derived group G2(2)'
Order = 6048 = 25.33.7.
Mult = 1.
Out = 2.
Standard generators of U3(3) are a and b where
a has order 2, b has order 6 and ab has order 7.
Standard generators of U3(3):2 = G2(2) are c
and d where c is in class 2B, d is in class 4D and
cd has order 7.
Presentations of U3(3) and U3(3):2 = G2(2) on their standard generators are given below.
< a, b | a2 = b6 = (ab)7 = [a, (ab2)3] = b3[b2, ab3a]2 = 1 >.
< c, d | c2 = d4 = (cd)7 = [c, d]6 = (cd(cd2)3)2 = [d2, cdc]3 = 1 >.
The representations of U3(3) available are:
- All primitive permutation representations.
-
Permutations on 28 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 36 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 63[a] points - on the cosets of 4.S4:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 63[b] points - on the cosets of 4^2:S3:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 2.
-
Dimension 6 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 14 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 32[a] over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 32[b] over GF(2) - the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
-
Dimension 3[b] over GF(9) - the natural representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 6[b] over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 7 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 15[b] over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 27 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 7 (up to Frobenius automorphisms).
-
Dimension 6 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 7 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 7[b] over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 14 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 21 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 21[b] over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 26 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 28[a] over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Some faithful irreducibles in characteristic 0
- Dimension 6 over Z[i]:
a and b (GAP).
- Dimension 7 over Z:
a and b (GAP).
- Dimension 7 over Z[i]:
a and b (GAP).
- Dimension 14 over Z (reducible over Z[i]):
a and b (GAP).
- Dimension 14 over Z:
a and b (GAP).
- Dimension 21 over Z:
a and b (GAP).
- Dimension 21 over Z[i]:
a and b (GAP).
- Dimension 42 over Z (reducible over Z[i]):
a and b (GAP).
- Dimension 27 over Z:
a and b (GAP).
- Dimension 28 over Z[i]:
a and b (GAP).
- Dimension 56 over Z (reducible over Z[i]):
a and b (GAP).
- Dimension 64 over Z (reducible over Q(b7)):
a and b (GAP).
The representations of U3(3):2 = G2(2) available are:
-
Permutations on 63 points - on the cosets of 4^2:D12:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- All faithful irreducibles in characteristic 2.
-
Dimension 6 over GF(2) - exhibiting the isomorphism with G2(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 14 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 64 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 characters.
-
Dimension 6 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 12 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 7 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 30 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 27 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The maximal subgroups of U3(3) are as follows.
-
31+2:8, with generators
here.
-
L2(7), with generators
here.
-
4.S4.
-
42:S3, with generators
here.
The maximal subgroups of U3(3):2 = G2(2) are as follows.
-
U3(3), with standard generators
dd, cdcddd.
-
31+2:8:2.
-
L2(7):2.
-
4.S4:2.
-
42:D12.
At the moment, we are only going to give enough class representatives so that
we can sort out some of our generality problems.
Representatives of some of the 14 conjugacy classes of U3(3) are given below.
- 1A: identity [or a2].
- 2A: a.
- 3A: b2.
- 3B: abab-1 or [a, b].
- 4A: (ab-2)3.
- 4B: (ab2)3.
- 4C: ab3.
- 6A: b.
- 7A: ab.
- 7B: ab-1.
- 8A: ab2ab-1.
- 8B: abab-2.
- 12A: ab2.
- 12B: ab-2.
Representatives of some of the 16 conjugacy classes of U3(3):2 are given below.
- 1A: identity [or c2].
- 2A: d2.
- 3A: .
- 3B: .
- 4AB: .
- 4C: .
- 6A: .
- 7AB: cd.
- 8AB: .
- 12AB: .
- 2B: c.
- 4C: d.
- 6B: .
- 8C: .
- 12C: .
- 12D: .
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old U3(3) page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 19th July 2000.
Last updated 03.03.04 by SJN.
Information checked to
Level 0 on 19.07.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.