ATLAS: Exceptional group G2(4)
Order = 251596800.
Mult = 2.
Out = 2.
Standard generators
Standard generators of G2(4) are
a
and b where
a is in class 2A, b is in class 5C/D,
and ab has order 13.
Standard generators of the double cover 2G2(4) are pre-images
A
and B where B has order 5
and AB has order 13.
Standard generators of G2(4):2 are
c
and d where
c is in class 2C, d is in class 4D,
cd has order 13, and cdcddd has order 6.
Standard generators of the double cover 2.G2(4):2 are pre-images
C
and D where CD has order 13.
Automorphisms
The outer automorphism of G_{2}(4) can be realised by
the map taking (a,b) to (a,b^{2}).
Black box algorithms
Representations
The representations of G2(4) available are
- a and
b as
6 x 6 matrices over GF(4) - the natural representation.
- Some representations in characteristic 3.
- a and
b as
64 x 64 matrices over GF(3).
- a and
b as
78 x 78 matrices over GF(3).
- a and
b as
286 x 286 matrices over GF(3).
- Some representations in characteristic 5.
- a and
b as
65 x 65 matrices over GF(5).
- a and
b as
78 x 78 matrices over GF(5).
- a and
b as
350 x 350 matrices over GF(5).
- a and
b as
363 x 363 matrices over GF(5).
- a and
b as
650 x 650 matrices over GF(5).
- a and
b as
65 x 65 matrices over GF(7).
- a and
b as
78 x 78 matrices over GF(7).
- a and
b as
65 x 65 matrices over GF(13).
- a and
b as
78 x 78 matrices over GF(13).
- Some primitive permutation representations.
- a and
b as
permutations on 416 points.
- a and
b as
permutations on 1365 points - the first one in the Atlas.
- a and
b as
permutations on 1365 points - the second one in the Atlas.
- a and
b as
permutations on 2016 points.
- a and
b as
permutations on 2080 points.
- a and
b as
permutations on 20800 points.
The representations of G2(4):2 available are
- Some representations in characteristic 2.
- c and
d as
12 x 12 matrices over GF(2).
- c and
d as
28 x 28 matrices over GF(2).
- c and
d as
36 x 36 matrices over GF(2).
- c and
d as
128 x 128 matrices over GF(2).
- c and
d as
168 x 168 matrices over GF(2).
- c and
d as
196 x 196 matrices over GF(2).
- c and
d as
64 x 64 matrices over GF(3).
- c and
d as
65 x 65 matrices over GF(5).
- c and
d as
65 x 65 matrices over GF(7).
- c and
d as
65 x 65 matrices over GF(13).
The representations of 2.G2(4) available are
- A and
B as
12 x 12 matrices over GF(3).
- A and
B as
12 x 12 matrices over GF(5).
- A and
B as
92 x 92 matrices over GF(5).
- A and
B as
12 x 12 matrices over GF(7).
- A and
B as
12 x 12 matrices over GF(13).
The representations of 2.G2(4):2 available are
- C and
D as
12 x 12 matrices over GF(3) - isoclinic to the Atlas group.
- C and
D as
12 x 12 matrices over GF(25) - the Atlas group.
- C and
D as
12 x 12 matrices over GF(7).
- C and
D as
12 x 12 matrices over GF(169).
Maximal subgroups
The maximal subgroups of G2(4) are
- J2, with standard generators
here.
- 2^2+8:(A5 x 3), with generators
here (mapping to standard generators of A5).
- 2^4+6:(A5 x 3)
- U3(4):2, with standard generators
here.
- 3.L3(4):2, with (nonstandard) generators
here.
- U3(3):2, with (nonstandard) generators
here.
- A5 x A5
- L2(13), with standard generators
here.
The maximal subgroups of G2(4):2 are
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Last updated 24.03.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk