ATLAS: Exceptional group G_{2}(5)
Order = 5859000000 = 2^{6}.3^{3}.5^{6}.7.31.
Mult = 1.
Out = 1.
Type I standard generators of G_{2}(5) are a and b where
a has order 2 (is in class 2A), b is in class 3B, ab has order 7 and
ababb has order 15.
Type II standard generators of G_{2}(5) are x and y where
x has order 2 (is in class 2A), y is in class 5B and xy has order 7.
Without loss of generality, we can obtain (a, b) as
(a, b) = (x, ((xyxyxy^{4})^{4})^{(xy2)23(xy)5}).
Conversely the pair (X, Y) = (a, (babab)^{3}) is conjugate
(in G_{2}(5)) to (x, y).
In fact, (x, y) = (X, Y^{c}) = (X^{c}, Y^{c}),
where c = de^{4}dede^{3}dede^{2}d,
with d = [X, YXY^{2}]^{6} and
e = YXYXY[X, YXYXY]^{7}.
The representations of G_{2}(5) available are:
 Primitive permutation representations.

Permutations on 3906 points  on the cosets of N(5A):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 3906 points  on the cosets of N(5A^{2}):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 7750 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 7875 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 124 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 124 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 651 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Faithful irreducibles in characteristic 5 (up to dimension 1000).

Dimension 7 over GF(5)  the natural representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(5)  the Lie algebra:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 97 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 182 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 189 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 196 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 371 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 469 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 483 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 721 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 792 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The maximal subgroups of G_{2}(5) are as follows.
A set of generators for the maximal cyclic subgroups
can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Go to main ATLAS (version 2.0) page.
Go to exceptional groups page.
Go to old G2(5) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 12th May 1999.
Last updated 03.12.08 by JNB.
Information checked to
Level 0 on 12.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.