ATLAS: Fischer group Fi23
Order = 4089470473293004800.
Mult = 1.
Out = 1.
Standard generators
Standard generators of the Fischer group Fi23 are a
and b where
a is in class 2B,
b is in class 3D,
and ab has order 28.
Black box algorithms
To find standard generators for F23:
- Find any element of order 20, 28 or 60. It powers up to a 2B-element 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 F23 have been obtained.
- If you fail, then y is probably in the wrong conjugacy class.
Representations
The representations of Fi23 available are:
- a and
b as
permutations on 31671 points.
- a and
b as
782 × 782 matrices over GF(2).
- a and
b as
253 × 253 matrices over GF(3).
Maximal subgroups
The maximal subgroups of Fi23 are:
-
2.F22, with standard generators
here and (nonstandard) generators
a, abab(abb)^4ab.
-
O8+(3):S3, with generators
[a, babab]^2, (ab)^8(abb)^3(ab)^11.
-
2^2.U6(2).2, with generators
a, (b(ab)^12)^2.
-
S8(2), with generators
a, abab(abb)^3ab.
-
O7(3) x S3, with generators
[a, babab]^2, abababb(ababbabb)^2abababb.
-
2^11.M23, with generators
a, (ab)^7(ba)^2.
-
3^1+8.2^1+6.3^1+2.2S4
-
[3^10].(L3(3) x 2)
-
S12, with generators
a, (ab)^9(abb)^12(ab)^9.
-
(2^2 x 2^1+8).( 3 x U4(2)).2, with generators
[a, babab]^2, ab(abb)^13(ab)^11(abb)^13.
-
2^6+8:(A7 x S3)
-
S6(2) x S4
-
S4(4):4, with generators
a, ab(ba)^12(ab)^16(ba)^13.
-
L2(23), with standard generators
[a, babab]^2, b^((ab)^3(abb)^10ab)
Return to main ATLAS page.
Last updated 29.10.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk
jnb@for.mat.bham.ac.uk