ATLAS: Exceptional group ²F₄(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 which have been scanning these pages have misinterpreted the occurrence of this name as an indication of quite a different content on these pages from that which actually pertains.

Sorry.

Order = 17971200 = 2¹¹.3³.5².13.
Mult = 1.
Out = 2.

The following information is available for ²F₄(2)':

Standard generators.
Black box algorithms.
Presentations.
Representations.
Maximal subgroups.
Conjugacy classes.

Standard generators

Standard generators of the group ²F₄(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 ²F₄(2) = ²F₄(2)'.2 are c and d where c is in class 2A, d is in class 4F, cd has order 12 and cdcd²cd³ has order 4.

A pair of elements automorphic to (a, b) may be obtained as a' = c, b' = (cdcdcdcd)^dcddcdcddd.

Black box algorithms

To find standard generators for ²F₄(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 ²F₄(2) = ²F₄(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

Presentations for ²F₄(2)' and ²F₄(2)'.2 (respectively) on their standard generators are given below.

< a, b | a² = b³ = (ab)¹³ = [a, b]⁵ = [a, bab]⁴ = (ababababab^-1)⁶ = 1 >.

< c, d | c² = d⁴ = (cd)¹² = [c, d]⁵ = ((cd²)³cd)⁴d^-2 = [c, (dc)³d^-1(cd^-1cd)²d] = [c, dcdcd^-1cd²]² = (cd)⁴cd²cd(cd^-1)⁴cd(cd²cd^-1)²cdcd²cdcd(cd^-1)²cdcd² = 1 >.

These presentations are available in Magma format as 2F4(2)' on a and b and 2F4(2)'.2 on c and d.

Representations

The representations of ²F₄(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 ²F₄(2) = ²F₄(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).

Maximal subgroups

The maximal subgroups of ²F₄(2)' are:

L₃(3):2, with generators here.
L₃(3):2, with generators here.
2.[2⁸].5.4, with generators here.
L₂(25), with generators here.
2².[2⁸].S₃, with generators here.
A₆.2², with generators here.
A₆.2², with generators here.
5²:4A₄, with generators here.

The maximal subgroups of ²F₄(2) = ²F₄(2)'.2are:

²F₄(2)', with standard generators c, (cdcdcdcd)^(dcddcdcddd).
13:12 = F₁₅₆.
3¹⁺²:SD₁₆.
2.[2⁹].5.4.
L₂(25).2₃.
2².[2⁹].S₃.
5²:4S₄.

Conjugacy classes

Representatives of the 22 conjugacy classes ²F₄(2)' are given below.

1A: identity [or a²].
2A: a.
2B: [a, bab]².
3A: b.
4A: (ababab²)³.
4B: ababababab²abab²ab².
4C: ababab²ab² or [a, bab].
5A: abab² or [a, b].
6A: ababababab² or (ababab²)².
8A: (ab)⁵ab²(ababab²)²ab².
8B: (ab)⁵ab²ab²(ababab²)².
8C: abababab².
8D: abab(abababab²)² or (ab)⁶(ab²)⁵.
10A: abababab²abab²ab².
12A: ababab².
12B: abababab²ab².
13A: ab.
13B: (ab)².
16A: (ab)⁵ab²ab(ab²)³.
16B: (ab)⁵(ab²)³abab².
16C: (ab)³(ab²)³ababab².
16D: abababab²abab(ab²)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 ²F₄(2) = ²F₄(2)'.2 are given below.

1A: identity [or c²].
2A: c.
2B: d².
3A: (cd)⁴.
4A: [d², cdcdc] or (cdcdcdcd²)².
4B: (cd)⁵cd^-1cd² or (cdcd²)⁴.
4C: (cd²)² or [c, d²].
5A: cdcd^-1 or [c, d].
6A: (cd)².
8A: (cd²cd^-1)².
8B: (cdcd²)².
8CD: cd².
10A: cdcd^-1cd².
12AB: cdcd²cd^-1cd² or [d², cdc].
13AB: cdcdcd²cd².
16AC: ab²ab^-1ab^-1.
16BD: ababab².
4D: (cdcdcd^-1)⁵.
4E: (cdcd^-1cd^-1)⁵.
4F: d.
4G: cdcd(cdcd^-1cd²)²cd^-1.
8E: cdcdcdcd²cdcd^-1.
8F: cdcdcdcd².
12C: cd^-1.
12D: cd.
16E: cd²cd^-1.
16F: cdcd².
20A: cdcdcd^-1.
20B: cdcd^-1cd^-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.

ATLAS: Exceptional group 2F4(2)'