Order = 10073444472 = 2^{3}.3^{9}.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, cdcd^{2} has order 14 and cdcdcdcd^{2}cdcd^{2}cd^{2} has order 9.
Representations
Representations of R(27)
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- Permutation representations:
Number of points ID Generators Description Link 19684 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 702 Std Details 3 GF(27) 7 a Std Details
Representations of R(27):3
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- Permutation representations:
Number of points ID Generators Description Link 19684 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 702 Std Details 2 GF(4) 741 Std Details 3 GF(3) 21 Std Details
Maximal subgroups
Maximal subgroups of R(27)
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
3^{3+3+3}:26 | Program: Generators | ||
2 × L_{2}(27) | Program: Generators | ||
L_{2}(8):3 | Program: Generators | ||
37:6 = F_{222} | Program: Generators | ||
(2^{2} × D_{14}):3 | Program: Generators | ||
19:6 = F_{114} | Program: Generators |
Maximal subgroups of R(27):3
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
R(27) | Program: Standard generators | ||
3^{3+3+3}:26:3 | Program: Generators | ||
2 × L_{2}(27):3 | |||
3 × L_{2}(8):3 | |||
37:18 = F_{666} | Program: Generators | ||
A_{4} × 7:6 | |||
19:18 = F_{342} |
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 |
Download words for class representatives.