# ATLAS: Orthogonal group O7(3)

Order = 4585351680 = 29.39.5.7.13.
Mult = 6.
Out = 2.

### Standard generators

Standard generators of O7(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.O7(3) are preimages A and B where B has order 7 and AB has order 13.
Standard generators of the triple cover 3.O7(3) are preimages A and B where A has order 2 and B has order 7.
Standard generators of the sextuple cover 6.O7(3) are preimages A and B where A has order and AB has order .

Standard generators of O7(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.O7(3):2 are preimages C and D where C has order and CD has order .
Standard generators of the triple cover 3O7(3):2 are preimages C and D where D has order 7.
Standard generators of the sixfold cover 6.O7(3):2 are preimages C and D where C has order and CD has order .

### Automorphisms

The outer automorphism of O7(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).