metacyclic, supersoluble, monomial
Aliases: C9⋊C24, C18.C12, C36.2C6, 3- 1+2⋊C8, C9⋊C8⋊C3, C2.(C9⋊C12), C32.(C3⋊C8), C4.2(C9⋊C6), (C3×C12).6S3, C12.10(C3×S3), C6.3(C3×Dic3), (C3×C6).2Dic3, (C2×3- 1+2).C4, (C4×3- 1+2).2C2, C3.3(C3×C3⋊C8), SmallGroup(216,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C9 — C9⋊C24 |
Generators and relations for C9⋊C24
G = < a,b | a9=b24=1, bab-1=a2 >
Smallest permutation representation of C9⋊C24
►On 72 points
Generators in S72
(1 42 62 19 34 70 12 26 54)(2 63 35 13 55 43 20 71 27)(3 36 56 21 28 64 14 44 72)(4 57 29 15 49 37 22 65 45)(5 30 50 23 46 58 16 38 66)(6 51 47 9 67 31 24 59 39)(7 48 68 17 40 52 10 32 60)(8 69 41 11 61 25 18 53 33)
(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 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)
G:=sub<Sym(72)| (1,42,62,19,34,70,12,26,54)(2,63,35,13,55,43,20,71,27)(3,36,56,21,28,64,14,44,72)(4,57,29,15,49,37,22,65,45)(5,30,50,23,46,58,16,38,66)(6,51,47,9,67,31,24,59,39)(7,48,68,17,40,52,10,32,60)(8,69,41,11,61,25,18,53,33), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)>;
G:=Group( (1,42,62,19,34,70,12,26,54)(2,63,35,13,55,43,20,71,27)(3,36,56,21,28,64,14,44,72)(4,57,29,15,49,37,22,65,45)(5,30,50,23,46,58,16,38,66)(6,51,47,9,67,31,24,59,39)(7,48,68,17,40,52,10,32,60)(8,69,41,11,61,25,18,53,33), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72) );
G=PermutationGroup([[(1,42,62,19,34,70,12,26,54),(2,63,35,13,55,43,20,71,27),(3,36,56,21,28,64,14,44,72),(4,57,29,15,49,37,22,65,45),(5,30,50,23,46,58,16,38,66),(6,51,47,9,67,31,24,59,39),(7,48,68,17,40,52,10,32,60),(8,69,41,11,61,25,18,53,33)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)]])
C9⋊C24 is a maximal subgroup of
C8×C9⋊C6 C72⋊C6 C36.C12 Dic18⋊C6 D36⋊C6 Dic18.C6 D36.C6
C9⋊C24 is a maximal quotient of C9⋊C48
40 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 9A | 9B | 9C | 12A | 12B | 12C | 12D | 12E | 12F | 18A | 18B | 18C | 24A | ··· | 24H | 36A | ··· | 36F |
| order | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 24 | ··· | 24 | 36 | ··· | 36 |
| size | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 2 | 3 | 3 | 9 | 9 | 9 | 9 | 6 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 9 | ··· | 9 | 6 | ··· | 6 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 |
| type | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C3 | C4 | C6 | C8 | C12 | C24 | S3 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | C3×C3⋊C8 | C9⋊C6 | C9⋊C12 | C9⋊C24 |
| kernel | C9⋊C24 | C4×3- 1+2 | C9⋊C8 | C2×3- 1+2 | C36 | 3- 1+2 | C18 | C9 | C3×C12 | C3×C6 | C12 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 |
Matrix representation of C9⋊C24 ►in GL8(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 14 | 20 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 59 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 54 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 54 | 59 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 19 |
| 0 | 0 | 0 | 0 | 5 | 19 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 68 | 0 | 0 |
G:=sub<GL(8,GF(73))| [0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0],[14,6,0,0,0,0,0,0,20,59,0,0,0,0,0,0,0,0,54,5,0,0,0,0,0,0,59,19,0,0,0,0,0,0,0,0,0,0,5,14,0,0,0,0,0,0,19,68,0,0,0,0,54,5,0,0,0,0,0,0,59,19,0,0] >;
C9⋊C24 in GAP, Magma, Sage, TeX
C_9\rtimes C_{24} % in TeX
G:=Group("C9:C24"); // GroupNames label
G:=SmallGroup(216,15);
// by ID
G=gap.SmallGroup(216,15);
# by ID
G:=PCGroup([6,-2,-3,-2,-2,-3,-3,36,50,3604,1450,208,5189]);
// Polycyclic
G:=Group<a,b|a^9=b^24=1,b*a*b^-1=a^2>;
// generators/relations
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