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

Porting notes

Porting incomplete.

Standard generators

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

Standard generators of R(27):3 are c, d where c has order 2, d is in class 3D/D', cd has order 21, cdcd2 has order 14 and cdcdcdcd2cdcd2cd2 has order 9.

Representations

Representations of R(27)

Representations of R(27):3

Maximal subgroups

Maximal subgroups of R(27)

Subgroup Order Index Programs/reps
33+3+3:26 Program: Generators
2 × L2(27) Program: Generators
L2(8):3 Program: Generators
37:6 = F222 Program: Generators
(22 × D14):3 Program: Generators
19:6 = F114 Program: Generators

Maximal subgroups of R(27):3

Subgroup Order Index Programs/reps
R(27) Program: Standard generators
33+3+3:26:3 Program: Generators
2 × L2(27):3
3 × L2(8):3
37:18 = F666 Program: Generators
A4 × 7:6
19:18 = F342

Notes

(R(27)) 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

Conjugacy classes of R(27)

Conjugacy class Centraliser order Power up Class rep(s)
1A 10 073 444 472 (ababababbabababbababbababababbabababbababb)3
2A 19 656 (ababababbabababbababb)3
3A 19 683 (ababababbabababbababbaabababbababb)3
3B 1 458 Omitted owing to length.
3C 1 458 ababababbabababbababbababababbabababbababb
6A 54 ababababbabababbababb
6B 54 (ababababbabababbababb)5
7A 28 abababbabbabababbabbabababbabababbabbabababbabbabababb
9A 81 ababababbabababbababbaabababbababb
9B 81 ababbababababbabababbababbabababbabbabababbabbabababb
9C 81 Omitted owing to length.
13A 26 Omitted owing to length.
13B 26 abababbabababbabababbabababb
13C 26 Omitted owing to length.
13D 26 abababbabababb
13E 26 Omitted owing to length.
13F 26 abababbabababbabababbabababbabababbabababbabababbabababb
14A 28 abababbabbabababbabbabababb
14B 28 (abababbabbabababbabbabababb)3
14C 28 (abababbabbabababbabbabababb)9
19A 19 abab
19B 19 abababab
19C 19 ab
26A 26 (abababb)3
26B 26 (abababb)9
26C 26 abababb
26D 26 (abababb)63
26E 26 (abababb)7
26F 26 (abababb)21
37A 37 Omitted owing to length.
37B 37 Omitted owing to length.
37C 37 abbabababbabbabababb
37D 37 abbabababb
37E 37 abbabababbabbabababbabbabababbabbabababb
37F 37 Omitted owing to length.

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Conjugacy classes of R(27):3

Conjugacy class Centraliser order Power up Class rep(s)
1A 30 220 333 416
2A 58 968
3A 59 049
3B 4 374
3C 4 374
6A 162
6B 162
7A 84
9A 243 cddcdcdcddcdcddcddcdcddcdcdcddcdcdcddcdcddcdd
9B 243
9C 243
13A 26
13B 26
14A 28 cdcdd
19A 19 dcdcdcddcdcdcddcdcdd
26A 26 cdcdcddcdcddcdd
26B 26
37A 37 cdcdcdcddcdcddcdcdcdd
37B 37
3D 4 536
3E 4 536
6C 72 cdcddcdcdcddcdcdcddcdcddcdd
6D 72
9D 81 cddcdcdcddcdcdcddcdcdd
9E 81
9F 54
9G 54
9H 54
9I 54
18A 18 cdcdcdd
18B 18 (cdcdcddcdcdcddcdcdd)−1
18C 18
18D 18
21A 21 cd
21B 21
27A 27
27B 27 cdcdcddcdcdd
27C 27
27D 27
27E 27
27F 27

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