ATLAS: Exceptional group G2(3)

Order = 4245696 = 26.36.7.13.
Mult = 3.
Out = 2.

Standard generators

Standard generators of G2(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.G2(3) are preimages A and B where A has order 2 and AB has order 13.

Standard generators of G2(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.G2(3):2 are preimages C and D where CD has order 13.


Automorphisms

The outer automorphism of G2(3) can be realised by mapping (a, b) to (a, (abb)-3b(abb)3).

Presentations

Presentations of G2(3) and G2(3):2 on their standard generators are given below.

< a, b | a2 = b3 = (ab)13 = [a, b]13 = abab[a, b]4(ab)3[a, bab]3 = (((ab)3ab-1)2(ab)2(ab-1)2)2 = 1 >.

< c, d | c2 = d4 = (cd)13 = (cdcd2cd2)2 = [c, dcdcdcd-1cdcd(cd-1)3cd(cd-1)2] = 1 >.


Representations

The representations of G2(3) available are: The representations of 3.G2(3) available are: The representations of G2(3):2 available are: The representation of 3.G2(3):2 available is:

Maximal subgroups

The maximal subgroups of G2(3) are: The maximal subgroups of G2(3):2 are:
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Last updated 13.04.00 by RAW
R.A.Wilson, R.A.Parker and J.N.Bray