ATLAS: Fischer group Fi_{23}
Order = 4089470473293004800 =
2^{18}.3^{13}.5^{2}.7.11.13.17.23.
Mult = 1.
Out = 1.
The following information is available for Fi_{23}:
Standard generators of the Fischer group Fi_{23} are a
and b where a is in class 2B, b is in class 3D and
ab has order 28.
Finding generators
To find standard generators for Fi_{23}:
 Find any element of order 20, 28 or 60. It powers up to a 2Belement x.
 Find any element of order 3, y say, (probably as a power of an element
of order 9 or 18).
 Find a conjugate a of x and a conjugate b of y
whose product has order 28.
 If you succeed, standard generators for Fi23 have been obtained.
 If you fail, then y is probably in the wrong conjugacy class.
This algorithm is available in computer readable format:
finder for Fi_{23}.
Checking generators
To check that elements x and y of Fi_{23}
are standard generators:
 Check o(x) = 2
 Check o(y) = 3
 Check o(xy) = 28
 Check o(x^{yy}(xy)^{14}) = 5
This algorithm is available in computer readable format:
checker for Fi_{23}.
The representations of Fi_{23} available are:

Permutations on 31671 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 137632 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Permutations on 275264 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
— imprimitive.

Dimension 782 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 1494 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
— kindly donated by Jürgen Müller.

Dimension 253 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 528 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 782 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 782 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 782 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 782 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 782 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 782 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
The maximal subgroups of Fi_{23} are:

2.Fi_{22}, with standard generators
here and (nonstandard) generators
a, abab(abb)^{4}ab.

O_{8}^{+}(3):S_{3}, with generators
[a, babab]^{2}, (ab)^{8}(abb)^{3}(ab)^{11}.

2^{2}.U_{6}(2).2, with generators
a, (b(ab)^{12})^{2}.

S_{8}(2), with generators
a, abab(abb)^{3}ab.

O_{7}(3) × S_{3}, with generators
[a, babab]^{2}, abababb(ababbabb)^{2}abababb.

2^{11}.M_{23}, with generators
a, (ab)^{7}(ba)^{2}.

3^{1+8}.2^{1+6}.3^{1+2}.2S_{4}.

[3^{10}].(L_{3}(3) × 2).

S_{12}, with generators
a, (ab)^{9}(abb)^{12}(ab)^{9}.

(2^{2} × 2^{1+8}).(3 × U_{4}(2)).2, with generators
[a, babab]^{2}, ab(abb)^{13}(ab)^{11}(abb)^{13}.

2^{6+8}:(A_{7} × S_{3}).

S_{6}(2) × S_{4}.

S_{4}(4):4, with generators
a, ab(ba)^{12}(ab)^{16}(ba)^{13}.

L_{2}(23), with standard generators
[a, babab]^{2}, b^((ab)^{3}(abb)^{10}ab).
A set of generators for the maximal cyclic subgroups can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Problems of algebraic conjugacy are not yet dealt with.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old Fi23 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 7th June 2000.
Last updated 16.05.06 by JNB.
Information checked to
Level 0 on 07.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.