ATLAS: Orthogonal group O8+(2)
Order = 174182400 = 2^{12}.3^{5}.5^{2}.7.
Mult = 2 × 2.
Out = S_{3}.
See also ATLAS of Finite Groups, pp85-87.
Standard generators
Standard generators of O8+(2) are a and b where
a is in class 2E, b is in class 5A, ab is in class 12F
(or 12G) and ababababbababbabb has order 8.
[Actually, b is allowed to be in classes 5B or 5C as they are
automorphic to class 5A, but then you must change the class you allow
ab to belong to. To ensure that ab is in the correct class
relative to the class of b, we just need to check that abb has
order 15.]
Standard generators of the double cover 2.O8+(2) are preimages A
and B where B has order 5, ABB has order 15
and B is in class +5A.
[In 2.O8+(2), class +5A is not automorphic to classes +5B or +5C. A condition equivalent to B being in class +5A is that ABABB has order 15.]
Standard generators of 2^{2}.O8+(2) are preimages A and
B where B has order 5 and ABB has order 15.
Standard generators of O8+(2):2 are c and d where c is
in class 2F, d is in class 10BC and cd has order 18.
[If you can't distinguish class 10BC from class 10A, then also add the
condition that cdd has order 30.]
Standard generators of any 2.O8+(2).2 are preimages C and D
where CD(CD^{4})^{3} has order 3.
Representations
The representations of O8+(2) available are
- Some primitive permutation representations.
- a and
b as
permutations on 120 points.
- a and
b as
permutations on 135 points.
- Some irreducible representations in characteristic 2.
- a and
b as
8 × 8 matrices over GF(2) - the natural representation.
- Some irreducible representations in characteristic 3.
- a and
b as
28 × 28 matrices over GF(3).
- a and
b as
35 × 35 matrices over GF(3).
- a and
b as
48 × 48 matrices over GF(3).
- Some irreducible representations in characteristic 5.
- a and
b as
28 × 28 matrices over GF(5).
- a and
b as
35 × 35 matrices over GF(5).
- a and
b as
50 × 50 matrices over GF(5).
- a and
b as
83 × 83 matrices over GF(5).
- Some irreducible representations in characteristic 7.
- a and
b as
28 × 28 matrices over GF(7).
- a and
b as
35 × 35 matrices over GF(7).
- a and
b as
50 × 50 matrices over GF(7).
- a and
b as
84 × 84 matrices over GF(7).
- Some faithful irreducibles in characteristic 0
- Dimension 28 over Z:
a and b (GAP).
- Dimension 35 over Z:
a and b (GAP).
- Dimension 50 over Z:
a and b (GAP).
- Dimension 84 over Z:
a and b (GAP).
- Dimension 175 over Z:
a and b (GAP).
- Dimension 210 over Z:
a and b (GAP).
The representation of 2.O8+(2) available is
- A and
B as
8 × 8 matrices over GF(3).
- A and
B as
8 × 8 matrices over GF(5).
- A and
B as
8 × 8 matrices over GF(7).
- A and
B as
permutations on 240 points.
The representation of 2.O8+(2):2 available is
- C and
D as
8 × 8 matrices over GF(3) - NOT in SGens.
The representation of 2^2.O8+(2):3 available is
- E and
F as
24 × 24 matrices over GF(3).
The representation of 2^2.O8+(2):S3 available is
- G and
H as
24 × 24 matrices over GF(3).
Return to main ATLAS page.
Last updated 25.08.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk