ATLAS: Unitary group U_{3}(3), Derived group G_{2}(2)'
Order = 6048 = 2^{5}.3^{3}.7.
Mult = 1.
Out = 2.
Standard generators
Standard generators of U_{3}(3) are a and b where
a has order 2, b has order 6 and ab has order 7.
Standard generators of U_{3}(3):2 = G_{2}(2) are c
and d where c is in class 2B, d is in class 4D and
cd has order 7.
Presentations
Presentations of U_{3}(3) and U_{3}(3):2 = G_{2}(2) on their standard generators are given below.
< a, b | a^{2} = b^{6} = (ab)^{7} = [a, (ab^{2})^{3}] = b^{3}[b^{2}, ab^{3}a]^{2} = 1 >.
< c, d | c^{2} = d^{4} = (cd)^{7} = [c, d]^{6} = (cd(cd^{2})^{3})^{2} = [d^{2}, cdc]^{3} = 1 >.
Representations
The representations of U_{3}(3) available are:
- All primitive permutation representations.
- a and
b as
permutations on 28 points.
- a and
b as
permutations on 36 points.
- a and
b as
permutations on 63 points (the cosets of 4.S4).
- a and
b as
permutations on 63 points (the cosets of 4^2:S3).
- All faithful irreducibles in characteristic 2.
- a and
b as
6 × 6 matrices over GF(2).
- a and
b as
14 × 14 matrices over GF(2).
- a and
b as
32 × 32 matrices over GF(2).
- a and
b as
32 × 32 matrices over GF(2) - the dual of the above.
- All faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
- a and
b as
3 × 3 matrices over GF(9) - the natural representation.
- a and
b as
6 × 6 matrices over GF(9).
- a and
b as
7 × 7 matrices over GF(3).
- a and
b as
15 × 15 matrices over GF(9).
- a and
b as
27 × 27 matrices over GF(3).
- All faithful irreducibles in characteristic 7 (up to Frobenius automorphisms).
- a and
b as
6 × 6 matrices over GF(7).
- a and
b as
7 × 7 matrices over GF(7).
- a and
b as
7 × 7 matrices over GF(49).
- a and
b as
14 × 14 matrices over GF(7).
- a and
b as
21 × 21 matrices over GF(7).
- a and
b as
21 × 21 matrices over GF(49).
- a and
b as
26 × 26 matrices over GF(7).
- a and
b as
28 × 28 matrices over GF(49).
The representations of U_{3}(3):2 = G_{2}(2) available are:
- c and
d as
permutations on 63 points (the cosets of 4^2:D12).
- All faithful irreducibles in characteristic 2.
- c and
d as
6 × 6 matrices over GF(2) - exhibiting the isomorphism with G2(2).
- c and
d as
14 × 14 matrices over GF(2).
- c and
d as
64 × 64 matrices over GF(2).
- All faithful irreducibles in characteristic 3 - up to tensoring with linear characters.
- c and
d as
6 × 6 matrices over GF(3).
- c and
d as
12 × 12 matrices over GF(3).
- c and
d as
7 × 7 matrices over GF(3).
- c and
d as
30 × 30 matrices over GF(3).
- c and
d as
27 × 27 matrices over GF(3).
Maximal subgroups
The maximal subgroups of U_{3}(3) are as follows.
- 3^{1+2}:8.
- L_{2}(7).
- 4.S_{4}.
- 4^{2}:S_{3}.
The maximal subgroups of U_{3}(3):2 = G_{2}(2) are as follows.
- U_{3}(3), with standard generators dd, cdcddd.
- 3^{1+2}:8:2.
- L_{2}(7):2.
- 4.S_{4}:2.
- 4^{2}:D_{12}.
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Last updated 1st December 1998,
R.A.Wilson, R.A.Parker and J.N.Bray