metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C24:5C4, C8:3Dic3, C6.4C42, C6.2M4(2), C3:C8:4C4, (C2xC8).8S3, C3:2(C8:C4), C4.21(C4xS3), (C2xC4).92D6, (C2xC24).15C2, C12.41(C2xC4), C2.4(C4xDic3), C2.2(C8:S3), C22.10(C4xS3), (C4xDic3).6C2, (C2xDic3).3C4, C4.13(C2xDic3), (C2xC12).106C22, (C2xC3:C8).10C2, (C2xC6).11(C2xC4), SmallGroup(96,22)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:C4
G = < a,b | a24=b4=1, bab-1=a5 >
Quotients: C1, C2, C4, C22, S3, C2xC4, Dic3, D6, C42, M4(2), C4xS3, C2xDic3, C8:C4, C8:S3, C4xDic3, C24:C4
Smallest permutation representation of C24:C4
►Regular action on 96 points
(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 31 81 67)(2 36 82 72)(3 41 83 53)(4 46 84 58)(5 27 85 63)(6 32 86 68)(7 37 87 49)(8 42 88 54)(9 47 89 59)(10 28 90 64)(11 33 91 69)(12 38 92 50)(13 43 93 55)(14 48 94 60)(15 29 95 65)(16 34 96 70)(17 39 73 51)(18 44 74 56)(19 25 75 61)(20 30 76 66)(21 35 77 71)(22 40 78 52)(23 45 79 57)(24 26 80 62)
G:=sub<Sym(96)| (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,31,81,67)(2,36,82,72)(3,41,83,53)(4,46,84,58)(5,27,85,63)(6,32,86,68)(7,37,87,49)(8,42,88,54)(9,47,89,59)(10,28,90,64)(11,33,91,69)(12,38,92,50)(13,43,93,55)(14,48,94,60)(15,29,95,65)(16,34,96,70)(17,39,73,51)(18,44,74,56)(19,25,75,61)(20,30,76,66)(21,35,77,71)(22,40,78,52)(23,45,79,57)(24,26,80,62)>;
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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,31,81,67)(2,36,82,72)(3,41,83,53)(4,46,84,58)(5,27,85,63)(6,32,86,68)(7,37,87,49)(8,42,88,54)(9,47,89,59)(10,28,90,64)(11,33,91,69)(12,38,92,50)(13,43,93,55)(14,48,94,60)(15,29,95,65)(16,34,96,70)(17,39,73,51)(18,44,74,56)(19,25,75,61)(20,30,76,66)(21,35,77,71)(22,40,78,52)(23,45,79,57)(24,26,80,62) );
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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,31,81,67),(2,36,82,72),(3,41,83,53),(4,46,84,58),(5,27,85,63),(6,32,86,68),(7,37,87,49),(8,42,88,54),(9,47,89,59),(10,28,90,64),(11,33,91,69),(12,38,92,50),(13,43,93,55),(14,48,94,60),(15,29,95,65),(16,34,96,70),(17,39,73,51),(18,44,74,56),(19,25,75,61),(20,30,76,66),(21,35,77,71),(22,40,78,52),(23,45,79,57),(24,26,80,62)]])
C24:C4 is a maximal subgroup of
C48:C4 C24:12Q8 C4xC8:S3 D6.C42 C24:Q8 S3xC8:C4 D6.4C42 Dic3.M4(2) C24:C4:C2 D6:2M4(2) C3:C8:26D4 D4.S3:C4 C4:C4.D6 C12:Q8:C2 D4:S3:C4 C3:Q16:C4 (C2xC8).D6 (C2xQ8).36D6 Q8:3(C4xS3) C42.27D6 C42.198D6 C42.202D6 C42.31D6 Dic12:9C4 C24:3Q8 D24:9C4 C24:4Q8 C24:C2:C4 D24:10C4 C12.12C42 C24:33D4 Dic3xM4(2) C12.7C42 C24:D4 C24.54D4 D8:Dic3 C24:11D4 SD16:Dic3 C24.31D4 C24:9D4 Q16:Dic3 C24.37D4 D8:4Dic3 C72:C4 C3:C8:Dic3 C2.Dic32 C24:Dic3 C30.22C42 C30.23C42 C120:13C4 C30.4C42 C30.M4(2) C24:F5
C24:C4 is a maximal quotient of
C42.279D6 C24:C8 C48:C4 (C2xC24):5C4 C72:C4 C3:C8:Dic3 C2.Dic32 C24:Dic3 C30.22C42 C30.23C42 C120:13C4 C30.4C42 C30.M4(2) C24:F5
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Dic3 | D6 | M4(2) | C4xS3 | C4xS3 | C8:S3 |
| kernel | C24:C4 | C2xC3:C8 | C4xDic3 | C2xC24 | C3:C8 | C24 | C2xDic3 | C2xC8 | C8 | C2xC4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 2 | 1 | 4 | 2 | 2 | 8 |
Matrix representation of C24:C4 ►in GL3(F73) generated by
| 27 | 0 | 0 |
| 0 | 8 | 65 |
| 0 | 8 | 16 |
| 27 | 0 | 0 |
| 0 | 54 | 68 |
| 0 | 14 | 19 |
G:=sub<GL(3,GF(73))| [27,0,0,0,8,8,0,65,16],[27,0,0,0,54,14,0,68,19] >;
C24:C4 in GAP, Magma, Sage, TeX
C_{24}\rtimes C_4 % in TeX
G:=Group("C24:C4"); // GroupNames label
G:=SmallGroup(96,22);
// by ID
G=gap.SmallGroup(96,22);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,55,69,2309]);
// Polycyclic
G:=Group<a,b|a^24=b^4=1,b*a*b^-1=a^5>;
// generators/relations
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