ATLAS: Ree group R(27)
Order = 10073444472 = 2^{3}.3^{9}.7.13.19.37.
Mult = 1.
Out = 3.
The following information is available for R(27):
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.
An outer automorphism may be obtained by mapping (a, b) to (a, b^{abababbabb}).
We take G2standard 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.
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 x_{1} and x_{2} of x such that x_{1}x_{2} has order 3 or 9. Then x_{1}x_{2} 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.
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).
The maximal subgroups of R(27) are:

3^{3+3+3}:26, with generators (abababbababb)^{11}a(abababbababb)^{11},
(ababababbababb)^{14}(ababb)(ababababbababb)^{14}.

2 × L_{2}(27), with generators
(ab)^{10}b(ab)^{9}, (abb)^{8}(abababbababababbababb)^{2}(abb)^{8}.

L_{2}(8):3, with generators
a, (abb)^{6}(ababababbabababbababb)^{2}(abb)^{6}.

37:6 = F_{222}, with generators (ab)^{3}babab, (abb)^{6}(abababbababababbababb)^{2}(abb)^{6}.

(2^{2} × D_{14}):3, with generators (ab)^{9}b(ab)^{8}, (abb)^{4}(abababbababababbababb)^{2}(abb)^{4}.

19:6 = F_{114}, with generators abababa(ab)^{3}, (abb)^{1}(abababbababababbababb)^{2}abb.
The maximal subgroups of R(27):3 are:
NB: Let S be a Sylow 3subgroup 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).
The 35 conjugacy classes of R(27) are roughly as follows:
 1A: identity.
 2A: a.
 3A: b.
 3B/C: (abababab^{2}ababab^{2}abab^{2})^{2}.
 6A/B: abababab^{2}ababab^{2}abab^{2}.
 7A: abababab^{2}abab^{2}ab^{2} or (ab)^{12}(ab^{2})^{3}.
 9A: (ab)^{9}(ab^{2})^{3} or (ab)^{9}(ab^{2})^{9} or
(ab)^{8}ab^{2}abab^{2}.
 9B/C: ab(abababab^{2})^{2}ab^{2}.
 13A/B/C/D/E/F: abab^{2} or [a, b].
 14A/B/C: (ab)^{6}ab^{2}.
 19A/B/C: ab.
 26A/B/C/D/E/F: ababab^{2}.
 37A/B/C/D/E/F: ababab^{2}ab^{2} 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.
Check  Date  By whom  Remarks 
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.