metabelian, supersoluble, monomial
Aliases: C32:C54, C33.1C18, C3:S3:C27, (C3xC27):1S3, C3.2(S3xC27), C32:C27:1C2, (C32xC9).1C6, C9.6(C32:C6), C32.13(S3xC9), C3.5(C32:C18), (C3xC3:S3).C9, (C9xC3:S3).C3, (C3xC9).48(C3xS3), SmallGroup(486,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C32:C54 |
Generators and relations for C32:C54
G = < a,b,c | a3=b3=c54=1, ab=ba, cac-1=a-1b-1, cbc-1=b-1 >
Quotients: C1, C2, C3, S3, C6, C9, C18, C3xS3, C27, C54, S3xC9, C32:C6, C32:C18, S3xC27, C32:C54
Smallest permutation representation of C32:C54
►On 54 points
Generators in S54
(2 20 38)(3 21 39)(5 41 23)(6 42 24)(8 26 44)(9 27 45)(11 47 29)(12 48 30)(14 32 50)(15 33 51)(17 53 35)(18 54 36)
(1 37 19)(2 20 38)(3 39 21)(4 22 40)(5 41 23)(6 24 42)(7 43 25)(8 26 44)(9 45 27)(10 28 46)(11 47 29)(12 30 48)(13 49 31)(14 32 50)(15 51 33)(16 34 52)(17 53 35)(18 36 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)
G:=sub<Sym(54)| (2,20,38)(3,21,39)(5,41,23)(6,42,24)(8,26,44)(9,27,45)(11,47,29)(12,48,30)(14,32,50)(15,33,51)(17,53,35)(18,54,36), (1,37,19)(2,20,38)(3,39,21)(4,22,40)(5,41,23)(6,24,42)(7,43,25)(8,26,44)(9,45,27)(10,28,46)(11,47,29)(12,30,48)(13,49,31)(14,32,50)(15,51,33)(16,34,52)(17,53,35)(18,36,54), (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)>;
G:=Group( (2,20,38)(3,21,39)(5,41,23)(6,42,24)(8,26,44)(9,27,45)(11,47,29)(12,48,30)(14,32,50)(15,33,51)(17,53,35)(18,54,36), (1,37,19)(2,20,38)(3,39,21)(4,22,40)(5,41,23)(6,24,42)(7,43,25)(8,26,44)(9,45,27)(10,28,46)(11,47,29)(12,30,48)(13,49,31)(14,32,50)(15,51,33)(16,34,52)(17,53,35)(18,36,54), (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) );
G=PermutationGroup([[(2,20,38),(3,21,39),(5,41,23),(6,42,24),(8,26,44),(9,27,45),(11,47,29),(12,48,30),(14,32,50),(15,33,51),(17,53,35),(18,54,36)], [(1,37,19),(2,20,38),(3,39,21),(4,22,40),(5,41,23),(6,24,42),(7,43,25),(8,26,44),(9,45,27),(10,28,46),(11,47,29),(12,30,48),(13,49,31),(14,32,50),(15,51,33),(16,34,52),(17,53,35),(18,36,54)], [(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)]])
90 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9L | 9M | ··· | 9R | 18A | ··· | 18F | 27A | ··· | 27R | 27S | ··· | 27AJ | 54A | ··· | 54R |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 | 27 | ··· | 27 | 27 | ··· | 27 | 54 | ··· | 54 |
| size | 1 | 9 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 9 | ··· | 9 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | + | |||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C27 | C54 | S3 | C3xS3 | S3xC9 | S3xC27 | C32:C6 | C32:C18 | C32:C54 |
| kernel | C32:C54 | C32:C27 | C9xC3:S3 | C32xC9 | C3xC3:S3 | C33 | C3:S3 | C32 | C3xC27 | C3xC9 | C32 | C3 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 1 | 2 | 6 | 18 | 1 | 2 | 6 |
Matrix representation of C32:C54 ►in GL6(F109)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 90 | 45 | 0 | 0 | 0 | 0 |
| 5 | 0 | 63 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 107 | 0 | 0 | 0 | 63 | 0 |
| 102 | 0 | 0 | 0 | 0 | 45 |
| 45 | 0 | 0 | 0 | 0 | 0 |
| 0 | 45 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 0 | 0 | 0 |
| 45 | 0 | 0 | 63 | 0 | 0 |
| 17 | 0 | 0 | 0 | 63 | 0 |
| 12 | 0 | 0 | 0 | 0 | 63 |
| 83 | 0 | 0 | 0 | 0 | 70 |
| 59 | 0 | 0 | 38 | 0 | 55 |
| 29 | 0 | 0 | 0 | 38 | 26 |
| 0 | 0 | 66 | 0 | 0 | 43 |
| 59 | 0 | 0 | 0 | 0 | 55 |
| 29 | 38 | 0 | 0 | 0 | 26 |
G:=sub<GL(6,GF(109))| [1,90,5,0,107,102,0,45,0,0,0,0,0,0,63,0,0,0,0,0,0,1,0,0,0,0,0,0,63,0,0,0,0,0,0,45],[45,0,0,45,17,12,0,45,0,0,0,0,0,0,45,0,0,0,0,0,0,63,0,0,0,0,0,0,63,0,0,0,0,0,0,63],[83,59,29,0,59,29,0,0,0,0,0,38,0,0,0,66,0,0,0,38,0,0,0,0,0,0,38,0,0,0,70,55,26,43,55,26] >;
C32:C54 in GAP, Magma, Sage, TeX
C_3^2\rtimes C_{54} % in TeX
G:=Group("C3^2:C54"); // GroupNames label
G:=SmallGroup(486,16);
// by ID
G=gap.SmallGroup(486,16);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,3244,3250,11669]);
// Polycyclic
G:=Group<a,b,c|a^3=b^3=c^54=1,a*b=b*a,c*a*c^-1=a^-1*b^-1,c*b*c^-1=b^-1>;
// generators/relations
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