ATLAS: Orthogonal group O10+(2)
Order = 23499295948800 = 220.35.52.7.17.31.
Mult = 1.
Out = 2.
The following information is available for O10+(2):
Standard generators of O10+(2) are a and
b where a is in class 2A, b is in class 20A and
ab has order 21.
Standard generators of O10+(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 O10+(2) can be taken to map
(a, b) to (a, b-1).
To find standard generators for O10+(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 O10+(2).
To find standard generators for O10+(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 O10+(2):2.
Presentations of O10+(2)
and O10+(2):2 on their standard generators are given
below:
< a, b | a2 = b20 =
(ab)21 = (ab2)17 = . . . = 1 >.
< c, d | c2 = d16 =
(cd)45 = [c, d]3 =
[c, d2]2 =
[c, d3]3 =
[c, d4]2 =
[c, d5]2 =
[c, d6]2 =
[c, d7]2 =
(cd8)4 =
(cd2cd2cd-1)9
= 1 >.
The relations (cd)45 = [c, d]3 = 1
in the O10+(2):2 presentation are redundant.
These presentations are available in Magma format as follows:
O10+(2):2 on c and d.
The representations of O10+(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 ½-spin representation.
-
Dimension 16b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- the other ½-spin 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 O10+(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 O10+(2) are as follows.
The maximal subgroups of O10+(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.