metacyclic, supersoluble, monomial
Aliases: C27:3C18, D27:2C9, C92.1S3, C27:2C9:C2, (C3xC27).C6, C9.4(S3xC9), (C3xD27).C3, C3.3(C9xD9), (C3xC9).1D9, C3.2(C27:C6), C32.13(C3xD9), (C3xC9).53(C3xS3), SmallGroup(486,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — C27:3C18 |
Generators and relations for C27:3C18
G = < a,b | a27=b18=1, bab-1=a17 >
Quotients: C1, C2, C3, S3, C6, C9, D9, C18, C3xS3, C3xD9, S3xC9, C9xD9, C27:C6, C27:3C18
Smallest permutation representation of C27:3C18
►On 54 points
Generators in S54
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27)(28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54)
(1 32 13 47 25 35 10 50 22 38 7 53 19 41 4 29 16 44)(2 40 23 46 17 52 11 31 5 37 26 43 20 49 14 28 8 34)(3 48 6 45 9 42 12 39 15 36 18 33 21 30 24 54 27 51)
G:=sub<Sym(54)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,32,13,47,25,35,10,50,22,38,7,53,19,41,4,29,16,44)(2,40,23,46,17,52,11,31,5,37,26,43,20,49,14,28,8,34)(3,48,6,45,9,42,12,39,15,36,18,33,21,30,24,54,27,51)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54), (1,32,13,47,25,35,10,50,22,38,7,53,19,41,4,29,16,44)(2,40,23,46,17,52,11,31,5,37,26,43,20,49,14,28,8,34)(3,48,6,45,9,42,12,39,15,36,18,33,21,30,24,54,27,51) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27),(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)], [(1,32,13,47,25,35,10,50,22,38,7,53,19,41,4,29,16,44),(2,40,23,46,17,52,11,31,5,37,26,43,20,49,14,28,8,34),(3,48,6,45,9,42,12,39,15,36,18,33,21,30,24,54,27,51)]])
63 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 9A | ··· | 9I | 9J | ··· | 9O | 9P | ··· | 9U | 18A | ··· | 18F | 27A | ··· | 27AA |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 27 | 27 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 27 | ··· | 27 | 6 | ··· | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | S3 | D9 | C3xS3 | S3xC9 | C3xD9 | C9xD9 | C27:C6 | C27:3C18 |
| kernel | C27:3C18 | C27:2C9 | C3xD27 | C3xC27 | D27 | C27 | C92 | C3xC9 | C3xC9 | C9 | C32 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 3 | 2 | 6 | 6 | 18 | 3 | 6 |
Matrix representation of C27:3C18 ►in GL6(F109)
| 100 | 98 | 0 | 0 | 0 | 0 |
| 64 | 9 | 27 | 0 | 0 | 0 |
| 99 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 44 |
| 0 | 0 | 0 | 40 | 0 | 73 |
| 0 | 0 | 0 | 14 | 105 | 49 |
| 0 | 0 | 0 | 66 | 0 | 0 |
| 0 | 0 | 0 | 87 | 27 | 0 |
| 0 | 0 | 0 | 21 | 0 | 16 |
| 66 | 0 | 0 | 0 | 0 | 0 |
| 87 | 27 | 0 | 0 | 0 | 0 |
| 21 | 0 | 16 | 0 | 0 | 0 |
G:=sub<GL(6,GF(109))| [100,64,99,0,0,0,98,9,15,0,0,0,0,27,0,0,0,0,0,0,0,60,40,14,0,0,0,0,0,105,0,0,0,44,73,49],[0,0,0,66,87,21,0,0,0,0,27,0,0,0,0,0,0,16,66,87,21,0,0,0,0,27,0,0,0,0,0,0,16,0,0,0] >;
C27:3C18 in GAP, Magma, Sage, TeX
C_{27}\rtimes_3C_{18} % in TeX
G:=Group("C27:3C18"); // GroupNames label
G:=SmallGroup(486,15);
// by ID
G=gap.SmallGroup(486,15);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,2163,2169,381,8104,208,11669]);
// Polycyclic
G:=Group<a,b|a^27=b^18=1,b*a*b^-1=a^17>;
// generators/relations
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