ATLAS: Fischer group Fi_{24}'
Order = 1255205709190661721292800 =
2^{21}.3^{16}.5^{2}.7^{3}.11.13.17.23.29.
Mult = 3.
Out = 2.
The following information is available for Fi_{24}':
Standard generators of the Fischer group Fi_{24}' are a
and b where a is in class 2A, b is in class 3E,
ab has order 29 and abababb has order 33.
Standard generators of the triple cover 3.Fi_{24}' are preimages
A and B where A has order 2 and AB has order 29.
Standard generators of the automorphism group Fi_{24} =
Fi_{24}':2 are c and d where c is in class 2C,
d is in class 8D and cd has order 29.
Standard generators of 3.Fi_{24}':2 are preimages
C and D where D has order 29.
A pair of generators conjugate to
a, b can be obtained as
a' = (cdd)^{10}, b' = (cdcdd)^{−10}(cdcdcddcdcdd)^{8}(cdcdd)^{10}.
Finding generators
To find standard generators for Fi_{24}':
 Find an element of order 22, 26, 28 or 60. It powers
up to a 2Aelement x.
 Find an element of order 39. It has a 5050 chance of
powering up to a 3Eelement y.
 Find conjugates a of x and b of y whose
product has order 29
[probability about 1 in 127].
 If you fail, you probably have a 3Aelement, so go back to
step 2.
 If you succeed, you definitely have a 3Eelement, so
keep going until abababb has order 33.
This algorithm is available in computer readable format:
finder for Fi_{24}'.
To find standard generators of Fi_{24} = Fi_{24}':2:
 Find an element of order 34, 46, 54, 70, or 78.
It powers to an element x in class 2C.
 Find an element of order 40.
It powers to an element y in class 8D.
 Find conjugates c of x and d of y such that
cd has order 29.
These are standard generators.
This algorithm is available in computer readable format:
finder for Fi_{24}':2.
Checking generators
To check that elements x and y of Fi_{24}'
are standard generators:
 Check o(x) = 2
 Check o(y) = 3
 Check o(xy) = 29
 Check o(xyxyxyy) = 23
 Let z = (xy)^{6}y
 Check o(z) = 60
 Check o(x(z^{30})^{xyxy}) = 5
This algorithm is available in computer readable format:
checker for Fi_{24}'.
To check that elements x and y of Fi_{24}':2
are standard generators:
 Check o(x) = 2
 Check o(y) = 8
 Check o(xy) = 29
 Let z = xyxy^{6}
 Check o(z) = 54
 Check o(xz^{27}) = 3
 Check o(xyy) = 20
This algorithm is available in computer readable format:
checker for Fi_{24}':2.
The representations of Fi_{24}' available are:

Dimension 3774 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 781 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
The representations of 3.Fi_{24}' available are:

Dimension 783 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Permutations on 920808 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).
The representations of Fi_{24} = Fi_{24}':2 available are:

Dimension 781 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Permutations on 306936 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
The representations of 3.Fi_{24}':2 available are:

Dimension 1566 over GF(2):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).

Permutations on 920808 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP),
C and D (Magma).
The maximal subgroups of the simple group Fi_{24}' are:
The maximal subgroups of the automorphism group Fi_{24} =
Fi_{24}':2 are:
 Fi_{24}', with standard generators
here.
 Fi_{23} × 2,
with generators here.
 (2 × 2.Fi_{22}):2,
with generators here.
 S_{3} × O_{8}^{+}(3):S_{3},
with generators here.
 O_{10}^{−}(2):2,
with generators here.
 3^{7}.O_{7}(3):2,
with generators here.
 3^{1+10}:(U_{5}(2):2 × 2),
with generators here.
 2^{12}.M_{24},
with generators here.
 (2 × 2^{2}.U_{6}(2)):S_{3},
with generators here.
 2^{1+12}:3_{1}.U_{4}(3).(2^{2})_{122},
with generators here.
 3^{2+4+8}.(S_{5} × 2S_{4}),
with generators here.
 [3^{13}]:(L_{3}(3) × 2 × 2),
with generators here.
 S_{4} × O_{8}^{+}(2):S_{3},
with generators here.
 2^{3+12}.(L_{3}(2) × S_{6}),
with generators here.
 2^{7+8}.(S_{3} × A_{8}),
with generators here.
 (G_{2}(3) × S_{3} × S_{3}).2,
with generators here.
 (S_{9} × S_{5}),
with generators here.
 L_{2}(8):3 × S_{6},
with generators here.
 S_{7} × 7:6,
with generators here.
 7^{1+2}:(6 × S_{3}).2,
with generators here.
 F_{812} = 29:28,
with generators here.
A set of generators for the maximal cyclic subgroups up to automorphism
can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements, or their images under an outer automorphism.
Problems of algebraic conjugacy are also not dealt with.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old Fi24' page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 16th June 2000.
Last updated 16.05.06 by JNB.
Information checked to
Level 0 on 16.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.