ATLAS: Exceptional group G_{2}(3)
Order = 4245696 = 2^{6}.3^{6}.7.13.
Mult = 3.
Out = 2.
Standard generators
Standard generators of G_{2}(3) are a and b where
a has order 2, b is in class 3C and ab has order 13.
Standard generators of the triple cover 3.G_{2}(3) are preimages
A and B where A has order 2 and AB has order 13.
Standard generators of G_{2}(3):2 are c and d where c has order 2 (and is in class 2B), d is in class 4C,
cd has order 13 and cdd has order 6.
Standard generators of 3.G_{2}(3):2 are preimages C and
D where CD has order 13.
Automorphisms
The outer automorphism of G_{2}(3) can be
realised by mapping (a, b) to
(a, (abb)^{-3}b(abb)^{3}).
Presentations
Presentations of G_{2}(3) and G_{2}(3):2 on their standard generators are given below.
< a, b | a^{2} = b^{3} = (ab)^{13} = [a, b]^{13} = abab[a, b]^{4}(ab)^{3}[a, bab]^{3} = (((ab)^{3}ab^{-1})^{2}(ab)^{2}(ab^{-1})^{2})^{2} = 1 >.
< c, d | c^{2} = d^{4} = (cd)^{13} = (cdcd^{2}cd^{2})^{2} = [c, dcdcdcd^{-1}cdcd(cd^{-1})^{3}cd(cd^{-1})^{2}] = 1 >.
Representations
The representations of G_{2}(3) available are:
- Some faithful irreducibles in characteristic 2.
- a and
b as
14 × 14 matrices over GF(2).
- a and
b as
64 × 64 matrices over GF(4).
- a and
b as
78 × 78 matrices over GF(2).
- a and
b as
90 × 90 matrices over GF(2).
- a and
b as
90 × 90 matrices over GF(2).
- a and
b as
90 × 90 matrices over GF(2).
- a and
b as
378 × 378 matrices over GF(2).
- All faithful irreducibles in characteristic 3.
- a and
b as
7 × 7 matrices over GF(3) - the natural representation.
- a and
b as
7 × 7 matrices over GF(3) - the image of the above under an outer automorphism.
- a and
b as
27 × 27 matrices over GF(3).
- a and
b as
27 × 27 matrices over GF(3).
- a and
b as
49 × 49 matrices over GF(3).
- a and
b as
189 × 189 matrices over GF(3).
- a and
b as
189 × 189 matrices over GF(3).
- a and
b as
729 × 729 matrices over GF(3) - the Steinberg representation.
- a and b as
14 × 14 matrices over Z.
The representations of 3.G_{2}(3) available are:
- A and
B as
27 × 27 matrices over GF(4).
- A and
B as
27 × 27 matrices over GF(7).
- A and
B as
27 × 27 matrices over GF(13).
- A and
B as
permutations on 1134 points.
The representations of G_{2}(3):2 available are:
- c and
d as
permutations on 756 points.
- c and
d as
14 × 14 matrices over GF(2).
- c and
d as
14 × 14 matrices over GF(3).
- c and
d as
14 × 14 matrices over GF(7).
- c and
d as
14 × 14 matrices over GF(13).
- c and d as
14 × 14 matrices over Z[i3].
The representation of 3.G_{2}(3):2 available is:
- C and
D as
54 × 54 matrices over GF(2).
Maximal subgroups
The maximal subgroups of G_{2}(3) are:
The maximal subgroups of G_{2}(3):2 are:
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Last updated 13.04.00 by RAW
R.A.Wilson, R.A.Parker and J.N.Bray