# ATLAS: Ree group R(27)

Order = 10073444472 = 23.39.7.13.19.37.
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 x1 and x2 of x such that x1x2 has order 3 or 9. Then x1x2 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
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: abababab2ababab2abab2.
• 7A: abababab2abab2ab2 or (ab)12(ab2)3.
• 9A: (ab)9(ab2)3 or (ab)9(ab2)9.
• 9B/C: .
• 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].