ATLAS: Ree group R(27)
Order = 10073444472 = 2^{3}.3^{9}.7.13.19.37.
Mult = 1.
Out = 3.
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 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'.
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).
- 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 3A-element, y say.
- Find conjugates a of x and b of y such that ab has order 19.
- 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.
- Find an element of order 21. Its seventh power, y say, is in class 3D (or 3D').
Representations
The representations of R(27) available are
- a and
b as
permutations on 19684 points.
- a and
b as
702 × 702 matrices over GF(2).
- a and
b as
7 × 7 matrices over GF(27) - the natural representation.
The representations of R(27):3 available are
- c and
d as
702 × 702 matrices over GF(2).
- c and
d as
21 × 21 matrices over GF(3).
Maximal subgroups
The maximal subgroups of R(27) are
- 3^3+6:26, 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.
- 37:6, with generators (ab)^-3babab, (abb)^-6(abababbababababbababb)^2(abb)^6.
- (2^2 × D14):3, with generators (ab)^-9b(ab)^8, (abb)^-4(abababbababababbababb)^2(abb)^4.
- 19:6, with generators abababa(ab)^-3, (abb)^-1(abababbababababbababb)^2abb.
The maximal subgroups of R(27):3 are
Conjugacy classes
The 35 conjugacy classes of R(27) are roughly as follows:
- 1A: identity.
- 2A: a.
- 3A: b.
- 3B/C: .
- 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}.
- 9B/C: .
- 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].
Return to main ATLAS page.
Last updated 17th October 1998,
R.A.Wilson, R.A.Parker and J.N.Bray