ATLAS: Exceptional group ^{2}F_{4}(2)'
An Apology
We apologise to the eminent mathematician whose name is
usually attached to this group for removing his name from this page
and those linked to or from it. The reason is that certain webcrawlers
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Order = 17971200 = 2^{11}.3^{3}.5^{2}.13.
Mult = 1.
Out = 2.
The following information is available for ^{2}F_{4}(2)':
Standard generators of the group ^{2}F_{4}(2)' are a
and b where a is in class 2A, b has order 3 and ab
has order 13.
Standard generators of its automorphism group ^{2}F_{4}(2) =
^{2}F_{4}(2)'.2 are c and d where c is in
class 2A, d is in class 4F, cd has order 12 and
cdcd^{2}cd^{3} has order 4.
A pair of elements automorphic to (a, b) may be obtained as
a' = c, b' = (cdcdcdcd)^{dcddcdcddd}.
To find standard generators for ^{2}F_{4}(2)':
 Find any element of order 10 or 16. This powers up to x
in class 2A.
[The probability of success at each attempt is 7 in 20 (about 1 in 3).]
 Find any element of order 3, 6 or 12. This powers up to y
in class 3A.
[The probability of success at each attempt is 7 in 27 (about 1 in 4).]
 Find a conjugate a of x and a conjugate b of
y, whose product has order 13.
[The probability of success at each attempt is 8 in 65 (about 1 in 8).]
To find standard generators for ^{2}F_{4}(2) =
^{2}F_{4}(2)'.2:
 Find any element of order 10, 16 or 20. This powers up to x
in class 2A.
[The probability of success at each attempt is 2 in 5 (about 1 in 3).]
 Find any outer element of order 12. This powers up to y
in class 4F.
Note that elements of order 20 are outer elements, and elements
of orders 1, 2, 3, 6, 10 or 13 are inner.
[1 in 3 outer elements have order 12.]
 Find a conjugate c of x and a conjugate d of
y, whose product has order 12, and such that cdcddcddd
has order 4.
[The probability of success at each attempt is 64 in 585 (about 1 in 9).]
Presentations for ^{2}F_{4}(2)' and
^{2}F_{4}(2)'.2 (respectively) on their standard generators
are given below.
< a, b  a^{2} = b^{3} = (ab)^{13} = [a, b]^{5} = [a, bab]^{4} = (ababababab^{1})^{6} = 1 >.
< c, d  c^{2} = d^{4} = (cd)^{12} = [c, d]^{5} = ((cd^{2})^{3}cd)^{4}d^{2} = [c, (dc)^{3}d^{1}(cd^{1}cd)^{2}d] = [c, dcdcd^{1}cd^{2}]^{2} =
(cd)^{4}cd^{2}cd(cd^{1})^{4}cd(cd^{2}cd^{1})^{2}cdcd^{2}cdcd(cd^{1})^{2}cdcd^{2} = 1 >.
These presentations are available in Magma format as
2F4(2)' on a and b and
2F4(2)'.2 on c and d.
The representations of ^{2}F_{4}(2)' available are:
 All faithful irreducibles in characteristic 2 up to automorphisms.

Dimension 26 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 246 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 2048 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary).
 Some representations in characteristic 3.

Dimension 26 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 26 over GF(3)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(9)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 54 over GF(3)  reducible over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

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

Dimension 124 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 124 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 5.

Dimension 26 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 26 over GF(25)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(5)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 52 over GF(5)  reducible over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

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

Dimension 109 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 109 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 218 over GF(5)  reducible over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 13.

Dimension 26 over GF(169):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(13)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 52 over GF(13)  reducible over GF(169):
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).
 All primitive permutation representations (up to automorphisms)

Permutations on 1600 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1755 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 2304 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 2925 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 12480 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 14976 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of ^{2}F_{4}(2) = ^{2}F_{4}(2)'.2 available are:
 Some representations in characteristic 2.

Dimension 26 over GF(2)  the natural representation:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 246 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some representations in characteristic 3.

Dimension 52 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 54 over GF(3):
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).
 Some representations in characteristic 5

Dimension 27 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 52 over GF(5):
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 218 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some representations in characteristic 13

Dimension 27 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 27 over GF(13)  the dual of the above:
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 1374 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary).
 Some permutation representations

Permutations on 1755 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 2304 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The maximal subgroups of ^{2}F_{4}(2)' are:
The maximal subgroups of ^{2}F_{4}(2) = ^{2}F_{4}(2)'.2are:
Representatives of the 22 conjugacy classes ^{2}F_{4}(2)' are
given below.
 1A: identity [or a^{2}].
 2A: a.^{ }
 2B: [a, bab]^{2}.
 3A: b.^{ }
 4A: (ababab^{2})^{3}.
 4B: ababababab^{2}abab^{2}ab^{2}.
 4C: ababab^{2}ab^{2} or [a, bab].
 5A: abab^{2} or [a, b].
 6A: ababababab^{2} or (ababab^{2})^{2}.
 8A: (ab)^{5}ab^{2}(ababab^{2})^{2}ab^{2}.
 8B: (ab)^{5}ab^{2}ab^{2}(ababab^{2})^{2}.
 8C: abababab^{2}.
 8D: abab(abababab^{2})^{2} or (ab)^{6}(ab^{2})^{5}.
 10A: abababab^{2}abab^{2}ab^{2}.
 12A: ababab^{2}.
 12B: abababab^{2}ab^{2}.
 13A: ab.^{ }
 13B: (ab)^{2}.
 16A: (ab)^{5}ab^{2}ab(ab^{2})^{3}.
 16B: (ab)^{5}(ab^{2})^{3}abab^{2}.
 16C: (ab)^{3}(ab^{2})^{3}ababab^{2}.
 16D: abababab^{2}abab(ab^{2})^{3}.
A program to calculate them is given here
and a program to calculate representatives of the maximal cyclic subgroups
is given here.
Representatives of the 29 conjugacy classes ^{2}F_{4}(2) = ^{2}F_{4}(2)'.2 are given below.
 1A: identity [or c^{2}].
 2A: c.^{ }
 2B: d^{2}.
 3A: (cd)^{4}.
 4A: [d^{2}, cdcdc] or (cdcdcdcd^{2})^{2}.
 4B: (cd)^{5}cd^{1}cd^{2} or (cdcd^{2})^{4}.
 4C: (cd^{2})^{2} or [c, d^{2}].
 5A: cdcd^{1} or [c, d].
 6A: (cd)^{2}.
 8A: (cd^{2}cd^{1})^{2}.
 8B: (cdcd^{2})^{2}.
 8CD: cd^{2}.
 10A: cdcd^{1}cd^{2}.
 12AB: cdcd^{2}cd^{1}cd^{2} or [d^{2}, cdc].
 13AB: cdcdcd^{2}cd^{2}.
 16AC: ab^{2}ab^{1}ab^{1}.
 16BD: ababab^{2}.
 4D: (cdcdcd^{1})^{5}.
 4E: (cdcd^{1}cd^{1})^{5}.
 4F: d.^{ }
 4G: cdcd(cdcd^{1}cd^{2})^{2}cd^{1}.
 8E: cdcdcdcd^{2}cdcd^{1}.
 8F: cdcdcdcd^{2}.
 12C: cd^{1}.
 12D: cd.^{ }
 16E: cd^{2}cd^{1}.
 16F: cdcd^{2}.
 20A: cdcdcd^{1}.
 20B: cdcd^{1}cd^{1}.
A program to calculate them is given here
and a program to calculate representatives of the maximal cyclic subgroups
is given here.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to exceptional groups page.
Go to old 2F4(2)' page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 23rd April 1999.
Last updated 03.03.03 by RAW.
Information checked to
Level 0 on 28.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.