ATLAS: Exceptional group G_{2}(3)
Order = 4245696 = 2^{6}.3^{6}.7.13.
Mult = 3.
Out = 2.
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 irreducible representaions in characteristic 2.

Dimension 14 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 90 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 90 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 90 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 378 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 7 over GF(3)  the natural representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 7 over GF(3)  the image of the above under an outer automorphism:
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).

Dimension 27 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 49 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 189 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 189 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 729 over GF(3)  the Steinberg representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

a and b as
14 × 14 matrices over Z.
The representations of 3.G_{2}(3) available are:

Dimension 27 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 27 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 27 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 1134 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of G_{2}(3):2 available are:

Dimension 14 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 434 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 756 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

c and d as
14 × 14 matrices over Z[i3].
The representation of 3.G_{2}(3):2 available is:

Dimension 54 over GF(2):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
Maximal subgroups
The maximal subgroups of G_{2}(3) are:
 U_{3}(3):2,
with generators here.
 U_{3}(3):2,
with generators here.
 (3^{2} × 3^{1+2}):2S_{4},
with generators here.
 (3^{2} × 3^{1+2}):2S_{4},
with generators here.
 L_{3}(3):2,
with generators here.
 L_{3}(3):2,
with generators here.
 L_{2}(8):3,
with generators here.
 2^{3}.L_{3}(2),
with generators here.
 L_{2}(13),
with generators here.
 2^{1+4}:3^{2}:2,
with generators here.
The maximal subgroups of G_{2}(3):2 are:
Conjugacy classes
A set of generators for the maximal cyclic subgroups of
G2(3)
can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
A set of generators for the maximal cyclic subgroups of
G2(3):2
can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
An outer automorphism of
G2(3)
can be obtained
by running this program on the standard
generators.
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Version 2.0 created on 28th April 2000.
Last updated 17.05.00 by JNB.
Information checked to
Level 0 on 28.04.00 by RAP.
R.A.Wilson, R.A.Parker and J.N.Bray.