ATLAS: Ree group R(27)

Order = 10073444472 = 2³.3⁹.7.13.19.37.
Mult = 1.
Out = 3.

The following information is available for R(27):

Standard generators.
Automorphisms.
Black box algorithms.
Representations.
Maximal subgroups.
Conjugacy classes.
Checks applied.

Standard generators

Standard generators of R(27) are a and b where a has order 2, b is in class 3A and ab has order 19.

Standard [G1-]generators of R(27):3 are c and d where c has order 2, d is in class 3D (or 3D'), cd has order 21, cdcdd has order 14 and cdcdcdcddcdcddcdd has order 9. These conditions distinguish classes 3D and 3D'.

NB: d is a conjugate of the Frobenius automorphism that cubes field elements.

Automorphisms

An outer automorphism may be obtained by mapping (a, b) to (a, b^abababbabb).
We take G2-standard generators of R(27):3 to be a, b and the above automorphism.
We may obtain (a conjugate of) (c, d) by setting c = a and d = v^avav where v = (abu)⁷ and u is the above automorphism.

Black box algorithms

To find standard generators of R(27):

Find an element x of order 2 (by taking a suitable power of any element of even order).
[The probability of success at each attempt is 3 in 8 (about 1 in 3).]
Find conjugates x₁ and x₂ of x such that x₁x₂ has order 3 or 9. Then x₁x₂ powers to a 3Aelement, y say.
[The probability of success at each attempt is 20384 in 512487 (about 1 in 25).]
As an alternative to the previous step, just find y as the cube of an element of order 9.
[The probability of success at each attempt is 1 in 27, only fractionally less than the above.]
Find conjugates a of x and b of y such that ab has order 19.
[The probability of success at each attempt is 81 in 703 (about 1 in 9).]
Now a and b are standard generators of R(27).

To find standard generators of R(27).3:

Find an element x of order 2 (by taking a suitable power of any element of even order).
[The probability of success at each attempt is 3 in 8 (about 1 in 3).]
Find an element of order 21. Its seventh power, y say, is in class 3D (or 3D').
[The probability of success at each attempt is 2 in 21 (about 1 in 11).]
Find conjugates c of x and d of y such that cd has order 21 and cdcdd has order 14.
[The probability of success at each attempt is 56 in 6327 (about 1 in 113).]
If cdcdcdcddcdcddcdd has order 21 then invert d.
Now cdcdcdcddcdcddcdd has order 9, and c and d are standard generators of R(27).3.

Representations

The representations of R(27) available are:

Dimension 7 over GF(27) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 702 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Permutations on 19684 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

The representations of R(27):3 available are:

Dimension 21 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 702 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Permutations on 19684 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

Maximal subgroups

The maximal subgroups of R(27) are:

3³⁺³⁺³:26, with generators (abababbababb)^-11a(abababbababb)¹¹, (ababababbababb)¹⁴(ababb)(ababababbababb)^-14.
2 × L₂(27), with generators (ab)^-10b(ab)⁹, (abb)^-8(abababbababababbababb)²(abb)⁸.
L₂(8):3, with generators a, (abb)^-6(ababababbabababbababb)²(abb)⁶.
37:6 = F₂₂₂, with generators (ab)^-3babab, (abb)^-6(abababbababababbababb)²(abb)⁶.
(2² × D₁₄):3, with generators (ab)^-9b(ab)⁸, (abb)^-4(abababbababababbababb)²(abb)⁴.
19:6 = F₁₁₄, with generators abababa(ab)^-3, (abb)^-1(abababbababababbababb)²abb.

The maximal subgroups of R(27):3 are:

R(27), with standard generators (cd)^-2dcd, (cdcddcdcdcdd)⁹.
3³⁺³⁺³:26:3, with generators here.
2 × L₂(27):3.
3 × L₂(8):3.
37:18 = F₆₆₆, with generators here.
A₄ × 7:6.
19:18 = F₃₄₂.

NB: Let S be a Sylow 3-subgroup of R(27). Then we have 1 < Z(S) < S' < S with |Z(S)| = 27 and |S'| = 729, Both Z(S) and S' are elementary abelian. The quotient S/Z(S) is special of exponent 3 and centre of order 27. All elements of S not in S' have order 9, and cube into Z(S).

Conjugacy classes

The 35 conjugacy classes of R(27) are roughly as follows:

1A: identity.
2A: a.
3A: b.
3B/C: (abababab²ababab²abab²)².
6A/B: abababab²ababab²abab².
7A: abababab²abab²ab² or (ab)¹²(ab²)³.
9A: (ab)⁹(ab²)³ or (ab)⁹(ab²)⁹ or (ab)⁸ab²abab².
9B/C: ab(abababab²)²ab².
13A/B/C/D/E/F: abab² or [a, b].
14A/B/C: (ab)⁶ab².
19A/B/C: ab.
26A/B/C/D/E/F: ababab².
37A/B/C/D/E/F: ababab²ab² or [a, bab].

A program to calculate representatives of the maximal cyclic subgroups of R(27) is given here.

A program to calculate representatives of the maximal cyclic subgroups of R(27):3 is given here.

Checks applied

Check	Date	By whom
Links work (except representations)
Links to (meataxe) representations work and have right degree and field
All info from v1 is included
HTML page standard
Word program syntax
Word programs applied
All necessary standard generators are defined	19.02.03	JNB
All representations are in standard generators

Go to main ATLAS (version 2.0) page.

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Go to old R(27) page - ATLAS version 1.

Anonymous ftp access is also available on for.mat.bham.ac.uk.

Version 2.0 created on 17th April 2000.
Last updated 19.02.03 by JNB.
Information checked to Level 0 on 17.04.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.