ATLAS: Orthogonal group O_{7}(3)
Order = 4585351680 = 2^{9}.3^{9}.5.7.13.
Mult = 6.
Out = 2.
Standard generators
Standard generators of O_{7}(3) are a and b where
a is in class 2A, b has order 7, ab has order 13
and abb has order 20.
Standard generators of the double cover 2.O_{7}(3) are preimages
A and B where B has order 7 and AB has order 13.
Standard generators of the triple cover 3.O_{7}(3) are preimages
A and B where A has order 2 and B has order 7.
Standard generators of the sextuple cover 6.O_{7}(3) are preimages
A and B where
A has order
and AB has order .
Standard generators of O_{7}(3):2 are
c
and d where
c is in class 2D,
d has order 7,
and cd has order 26,
and cdcdd has order 14.
Standard generators of the double cover 2.O_{7}(3):2 are preimages
C
and D where
C has order
and CD has order .
Standard generators of the triple cover 3O_{7}(3):2 are preimages
C
and D where
D has order 7.
Standard generators of the sixfold cover 6.O_{7}(3):2 are preimages
C
and D where
C has order
and CD has order .
Automorphisms
The outer automorphism of O_{7}(3) may be realised by mapping
(a, b) to (a, b^{-1}). This automorphism resides in
Class 2F.
Representations
The representations of O7(3) available are
- a and
b as
7 × 7 matrices over GF(3) - the natural representation.
- a and
b as
78 × 78 matrices over GF(2).
- a and
b as
permutations on 351 points.
- a and
b as
permutations on 364 points.
- a and
b as
permutations on 378 points.
- a and
b as
permutations on 1080 points.
The representations of O7(3):2 available are
- c and
d as
7 × 7 matrices over GF(3) - the natural representation.
- c and
d as
78 × 78 matrices over GF(2).
- c and
d as
permutations on 351 points.
The representations of 2.O7(3) available are
- A and
B as
8 × 8 matrices over GF(3) - the spin representation.
- A and
B as
permutations on 2160 points.
The representation of 2.O7(3).2 available is
- C and
D as
8 × 8 matrices over GF(3) - the spin representation.
The representation of 3.O7(3) available is
- A and
B as
27 × 27 matrices over GF(4).
The representation of 3.O7(3):2 available is
- C and
D as
54 × 54 matrices over GF(2).
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Last updated 2nd December 1998,
R.A.Wilson, R.A.Parker and J.N.Bray