ATLAS: Ree group R(27)

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

The following information is available for R(27):

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.


An outer automorphism may be obtained by mapping (a, b) to (a, babababbabb).
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 = vavav where v = (abu)7 and u is the above automorphism.

Black box algorithms

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


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

Maximal subgroups

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

CheckDateBy whomRemarks
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 defined19.02.03JNB
All representations are in standard generators

Main ATLAS page Go to main ATLAS (version 2.0) page.
Exceptional groups page Go to exceptional groups page.
Old R(27) page Go to old R(27) page - ATLAS version 1.
ftp access Anonymous ftp access is also available on

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.