metacyclic, supersoluble, monomial
Aliases: C9⋊C54, D9⋊C27, C92.C6, C9⋊C27⋊C2, (C3×D9).C9, (C9×D9).C3, C9.5(C9⋊C6), C3.3(S3×C27), (C3×C9).1C18, (C3×C27).1S3, C3.3(C9⋊C18), C32.14(S3×C9), (C3×C9).49(C3×S3), SmallGroup(486,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C54 |
Generators and relations for C9⋊C54
G = < a,b | a9=b54=1, bab-1=a2 >
Smallest permutation representation of C9⋊C54
►On 54 points
Generators in S54
(1 49 43 37 31 25 19 13 7)(2 44 32 20 8 50 38 26 14)(3 33 9 39 15 45 21 51 27)(4 10 16 22 28 34 40 46 52)(5 17 29 41 53 11 23 35 47)(6 30 54 24 48 18 42 12 36)
(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)| (1,49,43,37,31,25,19,13,7)(2,44,32,20,8,50,38,26,14)(3,33,9,39,15,45,21,51,27)(4,10,16,22,28,34,40,46,52)(5,17,29,41,53,11,23,35,47)(6,30,54,24,48,18,42,12,36), (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( (1,49,43,37,31,25,19,13,7)(2,44,32,20,8,50,38,26,14)(3,33,9,39,15,45,21,51,27)(4,10,16,22,28,34,40,46,52)(5,17,29,41,53,11,23,35,47)(6,30,54,24,48,18,42,12,36), (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([[(1,49,43,37,31,25,19,13,7),(2,44,32,20,8,50,38,26,14),(3,33,9,39,15,45,21,51,27),(4,10,16,22,28,34,40,46,52),(5,17,29,41,53,11,23,35,47),(6,30,54,24,48,18,42,12,36)], [(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 | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9L | 9M | ··· | 9U | 18A | ··· | 18F | 27A | ··· | 27R | 27S | ··· | 27AJ | 54A | ··· | 54R |
| order | 1 | 2 | 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 | 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 | C3×S3 | S3×C9 | S3×C27 | C9⋊C6 | C9⋊C18 | C9⋊C54 |
| kernel | C9⋊C54 | C9⋊C27 | C9×D9 | C92 | C3×D9 | C3×C9 | D9 | C9 | C3×C27 | C3×C9 | C32 | C3 | C9 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 1 | 2 | 6 | 18 | 1 | 2 | 6 |
Matrix representation of C9⋊C54 ►in GL6(𝔽109)
| 75 | 0 | 0 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 105 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 0 | 0 | 0 | 105 | 0 | 0 |
| 0 | 0 | 0 | 0 | 105 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 105 | 0 | 0 | 0 | 0 | 0 |
| 0 | 105 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(109))| [75,0,0,0,0,0,0,38,0,0,0,0,0,0,105,0,0,0,0,0,0,16,0,0,0,0,0,0,66,0,0,0,0,0,0,27],[0,0,0,0,105,0,0,0,0,0,0,105,0,0,0,27,0,0,0,105,0,0,0,0,0,0,105,0,0,0,27,0,0,0,0,0] >;
C9⋊C54 in GAP, Magma, Sage, TeX
C_9\rtimes C_{54} % in TeX
G:=Group("C9:C54"); // GroupNames label
G:=SmallGroup(486,30);
// by ID
G=gap.SmallGroup(486,30);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,43,68,8104,3250,208,11669]);
// Polycyclic
G:=Group<a,b|a^9=b^54=1,b*a*b^-1=a^2>;
// generators/relations
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