ATLAS: Orthogonal group O_{10}^{+}(2)
Order = 23499295948800 = 2^{20}.3^{5}.5^{2}.7.17.31.
Mult = 1.
Out = 2.
The following information is available for O_{10}^{+}(2):
Standard generators of O_{10}^{+}(2) are a and
b where a is in class 2A, b is in class 20A and
ab has order 21.
Standard generators of O_{10}^{+}(2):2 are c and
d where c is in class 2E, d has order 16 and
cd has order 45.
An outer automorphism of O_{10}^{+}(2) can be taken to map
(a, b) to (a, b^{1}).
To find standard generators for O_{10}^{+}(2):

Find any element of order 60. It powers up to x in class 2A and
y in class 20A.
[The probability of success at each attempt is 1 in 30.]

Find a conjugate a of x and a conjugate b of y such that ab has order 21.
[The probability of success at each attempt is 32 in 1581 (about 1 in 49).]

Now a and b are standard generators for O_{10}^{+}(2).
To find standard generators for O_{10}^{+}(2).2:

Find any element of order 34. It powers up to x in class 2E.
[The probability of success at each attempt is 1 in 17 (or 2 in 17 if you can restrict your search to outer elements only).]

Find any element, y say, of order 16.
[The probability of success at each attempt is 1 in 32 (or 1 in 16 if you can restrict your search to outer elements only).]

Find a conjugate c of x and a conjugate d of y such that cd has order 45.
[The probability of success at each attempt is 2 in 31 (about 1 in 16).]

Now c and d are standard generators for O_{10}^{+}(2):2.
Presentations of O_{10}^{+}(2)
and O_{10}^{+}(2):2 on their standard generators are given
below:
< a, b  a^{2} = b^{20} =
(ab)^{21} = (ab^{2})^{17} = . . . = 1 >.
< c, d  c^{2} = d^{16} =
(cd)^{45} = [c, d]^{3} =
[c, d^{2}]^{2} =
[c, d^{3}]^{3} =
[c, d^{4}]^{2} =
[c, d^{5}]^{2} =
[c, d^{6}]^{2} =
[c, d^{7}]^{2} =
(cd^{8})^{4} =
(cd^{2}cd^{2}cd^{1})^{9}
= 1 >.
The relations (cd)^{45} = [c, d]^{3} = 1
in the O_{10}^{+}(2):2 presentation are redundant.
These presentations are available in Magma format as follows:
O_{10}^{+}(2):2 on c and d.
The representations of O_{10}^{+}(2) available are:

Primitive permutation representations.

Permutations on 496 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 527 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 2295a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 2295b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 19840 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 23715 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 39680 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 imprimitive.

Faithful irreducibles in characteristic 2.

Dimension 10 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 16a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 a ½spin representation.

Dimension 16b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the other ½spin representation.

Dimension 44 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 100 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 144a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 in 10 × 16a.

Dimension 144b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 in 10 × 16b.

Dimension 164 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 320 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 416a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 in 44 × 16a.

Dimension 416b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 in 44 × 16b.

Dimension 670 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Faithful irreducibles in characteristic 3.

Dimension 155 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 185 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 868 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of O_{10}^{+}(2):2 available are:

Primitive permutation representations.

Permutations on 496 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 527 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 4590 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 imprimitive.

Faithful irreducibles in characteristic 2.

Dimension 10 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 the natural representation.

Dimension 32 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 the spin representation.

Dimension 44 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 100 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 164 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 288 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 320 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 670 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 832 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The maximal subgroups of O_{10}^{+}(2) are as follows.
The maximal subgroups of O_{10}^{+}(2):2 are as follows.
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old O10+(2) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 23rd January 2004.
Last updated 26.01.04 by JNB/SJN.
Information checked to
Level 0 on 13.10.03 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.