ATLAS: Fischer group Fi_{22}
Order = 64561751654400 = 2^{17}.3^{9}.5^{2}.7.11.13.
Mult = 6.
Out = 2.
The following information is available for Fi_{22}:
Standard generators of the Fischer group Fi_{22} are a and
b where a is in class 2A, b has order 13, ab has
order 11 and ababababbababbabb has order 12.
Standard generators of the double cover 2.Fi_{22} are
preimages A and
B where B has order 13 and AB has
order 11.
Standard generators of the triple cover 3.Fi_{22} are
preimages A and
B where A has order 2 and B has
order 13. The canonical central element is (AB)^{22}
The outer automorphism may be realised by mapping (a, b) to
a, (ab)^4bb(ab)^4.
Standard generators of the automorphism group Fi_{22}:2 are c
and d where c is in class 2A, d is in class 18E and
cd has order 42.
Standard generators of 2.Fi_{22}:2 are
preimages C and
D where (CD)^{5}D has order 30.
This is equivalent to saying D is in +18E and CD is in +42A.
Standard generators of 3.Fi_{22}:2 are
preimages C and
D where C has order 2.
Finding generators
To find standard generators for Fi_{22}:
 Find any element of order 14, 22 or 30. This then powers to a 2Aelement x.
 Find any element of order 13, y, say.
 Find a conjugate a of x and a conjugate b of y, whose product has order 11,
and such that (ab)^{2}(ababb)^{2}abb
has order 12. These are standard generators of Fi_{22}.
This algorithm is available in computer readable format:
finder for Fi_{22}.
To find standard generators for Fi_{22}.2:
 Find any element of order 22. Its 11th power is a 2Aelement x, say.
 Find an element y of order 42.
 Find a conjugate c of x and a conjugate z of y such that cz has order 18 and cz^{5} has order 42.
 Now c and d = cz are standard generators of Fi_{22}:2.
This algorithm is available in computer readable format:
finder for Fi_{22}.2.
Checking generators
To check that elements x and y of Fi_{22}
are standard generators:
 Check o(x) = 2
 Check o(y) = 13
 Check o(xy) = 11
 Check o(xyxyxyxyyxyxyyxyy) = 12
 Let z = xyxyyxyy
 Check o(z) = 30
 Check o(xz^{15}) = 3
This algorithm is available in computer readable format:
checker for Fi_{22}.
To check that elements x and y of Fi_{22}.2
are standard generators:
 Check o(x) = 2
 Check o(y) = 18
 Check o(xy) = 42
 Let z = xyxy^{5}xy^{4}
 Check o(z) = 22
 Check o(xz^{11}) = 3
 Check o((y^{9})^{xyyy}(xy)^{21}) = 3
This algorithm is available in computer readable format:
checker for Fi_{22}.2.
Presentations of Fi_{22} and Fi_{22}:2 in terms of their standard
generators are given below. (The relation (cd^{9})^{4} = 1 in the Fi_{22}:2 presentation is redundant.)
< a, b  a^{2} = b^{13} =
(ab)^{11} = (ab^{2})^{21} =
[a, b]^{3} =
[a, b^{2}]^{3} =
[a, b^{3}]^{3} =
[a, b^{4}]^{2} =
[a, b^{5}]^{3} =
[a, bab^{2}]^{3} =
[a, b^{1}ab^{2}]^{2} =
[a, bab^{5}]^{2} =
[a, b^{2}ab^{5}]^{2} = 1 >.
< c, d  c^{2} = d^{18} =
[c, d]^{3} =
[c, d^{2}]^{3} =
[c, d^{3}]^{3} =
[c, d^{4}]^{3} =
[c, d^{5}]^{3} =
[c, d^{6}]^{2} =
[c, d^{7}]^{2} =
[c, d^{8}]^{3} =
(cd^{9})^{4} =
[c, dcdcd^{2}cd] =
[c, d^{2}cd^{2}cd^{4}cd^{2}] =
((cd^{3})^{4}cd^{4})^{2} =
(cd^{4}cd^{5}cd^{5})^{5} =
(cdcd^{3})^{8} = 1 >.
These presentations are available in Magma format as follows:
Fi22 on a and b,
2.Fi22 on A and B,
Fi22:2 on c and d and
3.Fi22:2 on C and D.
Representations are available for groups isoclinic to one of the following:
[Actually, representations of 6.Fi22:2 are not yet available.]
The representations of Fi_{22} available are:
 Some primitive permutation representations:

Permutations on 3510 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 14080 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 61776 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 142155 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 694980 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations over GF(2):

Dimension 78 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 350 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 572 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducible representations over GF(3):

Dimension 77 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 351 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 924 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Also available in dimension 4823 over GF(3)  send email
for details.

Dimension 78 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 428 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 429 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 429 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 429 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.Fi_{22} available are:

Permutations on 28160 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 123552 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 176 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 352 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 352 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 352 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 352 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of 3.Fi_{22} available are:

Permutations on 185328 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 27 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 351 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of 6.Fi_{22} available are:

Permutations on 370656 points  kindly provided by Bernd Schröder.
A and
B (GAP).
The representations of Fi_{22}:2 available are:

Permutations on 3510 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 78 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 350 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 572 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 1352 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 77 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 351 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 924 over GF(3) (6/10/04  corrected to standard generators):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 78 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 428 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 78 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 429 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 78 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 429 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 78 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 429 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Dimension 78 over Z  kindly provided by Bernd Schröder.
c and
d (GAP).
The representations of 2.Fi_{22}:2 available are:

Dimension 352 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Permutations on 56320 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 2.Fi_{22}.4
(a group of shape
2.Fi22.4, in which outer `involutions' square to a scalar of order 4)
available are:

Dimension 352 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 3.Fi_{22}:2 available are:

Permutations on 185328 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 54 over GF(2):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 702 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 6.Fi_{22}:2 available are:
The maximal subgroups of Fi_{22} are:
 2.U6(2), with semistandard generators
(ab)^5a(ab)^5,
(abb)^3.
 O7(3), with standard generators
a,
(ababb)^5(abb)^3(ababb)^5.
 O7(3), with standard generators
here. (Nonstandard generators for a conjugate
are here.)
 O8+(2):S3, with generators
a, (bab)^3b^5.
 2^10:M22, with generators
bab^2ab^7, [a,b^4].
 2^6:S6(2), with generators
bab^11ab^6, [a,b^4].
 (2 × 2^1+8):(U4(2):2) =
2.2^1+8:(U4(2):2) with (nonstandard)
generators ab^4ab^4, bab^3ab^5
[The first 2^1+8 is extraspecial and the second one is elementary abelian.]
 U4(3):2 × S3, with generators
a(bab^1ab)^6, bab^6abab^5.
 ^2F4(2)', with standard generators
(abb)^3(abababb)^8(abb)^3,
(ababb)^1(babb)^3ababb.
 2^5+8:(S3 × A6), with generators
(ab)^3(ababb)^2(ab)^1,
(abb)^10(abababb)^4(abb)^10.
 3^1+6:2^3+4:3^2:2, with three generators
here.
 S10, with standard generators
(ab)^7a(ab)^7,
(abb)^5(babb)(abb)^5.
 S10, with standard generators
here. Shorter words for a conjugate are
here (also standard generators).
 M12, with standard generators
(abb)^4(ababababbababb)^6(abb)^4,
(ababb)^8(bababababbababbabb)^4(ababb)^8.
The maximal subgroups of Fi_{22}:2 are:
We add here some subgroups which may be useful for condensation
purposes:
A set of generators for the maximal cyclic subgroups of Fi_{22}
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.
A set of generators for the maximal cyclic subgroups of Fi_{22}:2
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 Fi22 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 17th January 2001.
Last updated 7.1.05 by SJN.
Information checked to
Level 0 on 17.01.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.