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)

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
19684StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(2)702StdDetails
3GF(27)7aStdDetails

### Representations of R(27):3

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
19684StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(2)702StdDetails
2GF(4)741StdDetails
3GF(3)21StdDetails

## 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)
1A10 073 444 472 (ababababbabababbababbababababbabababbababb)3
2A19 656 (ababababbabababbababb)3
3A19 683 (ababababbabababbababbaabababbababb)3
3B1 458 Omitted owing to length.
3C1 458 ababababbabababbababbababababbabababbababb
6A54 ababababbabababbababb
6B54 (ababababbabababbababb)5
7A28 abababbabbabababbabbabababbabababbabbabababbabbabababb
9A81 ababababbabababbababbaabababbababb
9B81 ababbababababbabababbababbabababbabbabababbabbabababb
9C81 Omitted owing to length.
13A26 Omitted owing to length.
13B26 abababbabababbabababbabababb
13C26 Omitted owing to length.
13D26 abababbabababb
13E26 Omitted owing to length.
13F26 abababbabababbabababbabababbabababbabababbabababbabababb
14A28 abababbabbabababbabbabababb
14B28 (abababbabbabababbabbabababb)3
14C28 (abababbabbabababbabbabababb)9
19A19 abab
19B19 abababab
19C19 ab
26A26 (abababb)3
26B26 (abababb)9
26C26 abababb
26D26 (abababb)63
26E26 (abababb)7
26F26 (abababb)21
37A37 Omitted owing to length.
37B37 Omitted owing to length.
37C37 abbabababbabbabababb
37D37 abbabababb
37E37 abbabababbabbabababbabbabababbabbabababb
37F37 Omitted owing to length.

### Conjugacy classes of R(27):3

Conjugacy class Centraliser order Power up Class rep(s)
1A30 220 333 416
2A58 968
3A59 049
3B4 374
3C4 374
6A162
6B162
7A84
9A243 cddcdcdcddcdcddcddcdcddcdcdcddcdcdcddcdcddcdd
9B243
9C243
13A26
13B26
14A28 cdcdd
19A19 dcdcdcddcdcdcddcdcdd
26A26 cdcdcddcdcddcdd
26B26
37A37 cdcdcdcddcdcddcdcdcdd
37B37
3D4 536
3E4 536
6C72 cdcddcdcdcddcdcdcddcdcddcdd
6D72
9D81 cddcdcdcddcdcdcddcdcdd
9E81
9F54
9G54
9H54
9I54
18A18 cdcdcdd
18B18 (cdcdcddcdcdcddcdcdd)−1
18C18
18D18
21A21 cd
21B21
27A27
27B27 cdcdcddcdcdd
27C27
27D27
27E27
27F27