metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:C8:7C2, C24:1C4:9C2, (C2xC8).11D6, D4:C4:7S3, C4:C4.138D6, (C2xD4).30D6, D6:3D4.6C2, C6.25(C4oD8), C12.Q8:7C2, C4.54(C4oD12), D4:Dic3:10C2, C2.12(Q8:3D6), C6.56(C8:C22), (C2xC24).11C22, (C6xD4).45C22, (C22xS3).16D4, C22.182(S3xD4), C12.152(C4oD4), C4.81(D4:2S3), C2.11(D8:3S3), (C2xC12).224C23, (C2xDic3).144D4, C3:3(C23.19D4), C4:Dic3.76C22, C2.15(C23.9D6), C6.23(C22.D4), C4:C4:7S3:4C2, (C3xD4:C4):7C2, (C2xC6).237(C2xD4), (C2xC3:C8).22C22, (S3xC2xC4).16C22, (C3xC4:C4).25C22, (C2xC4).331(C22xS3), SmallGroup(192,343)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24:1C4:C2
G = < a,b,c | a24=b4=c2=1, bab-1=a-1, cac=a19b2, cbc=b-1 >
Subgroups: 328 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, C23, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C22xC4, C2xD4, C2xD4, C3:C8, C24, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C22:C8, D4:C4, D4:C4, C4.Q8, C2.D8, C42:C2, C4:D4, C2xC3:C8, C4xDic3, C4:Dic3, D6:C4, C6.D4, C3xC4:C4, C2xC24, S3xC2xC4, C2xC3:D4, C6xD4, C23.19D4, C12.Q8, C24:1C4, D6:C8, D4:Dic3, C3xD4:C4, C4:C4:7S3, D6:3D4, C24:1C4:C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C22xS3, C22.D4, C4oD8, C8:C22, C4oD12, S3xD4, D4:2S3, C23.19D4, C23.9D6, D8:3S3, Q8:3D6, C24:1C4:C2
►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 52 25 74)(2 51 26 73)(3 50 27 96)(4 49 28 95)(5 72 29 94)(6 71 30 93)(7 70 31 92)(8 69 32 91)(9 68 33 90)(10 67 34 89)(11 66 35 88)(12 65 36 87)(13 64 37 86)(14 63 38 85)(15 62 39 84)(16 61 40 83)(17 60 41 82)(18 59 42 81)(19 58 43 80)(20 57 44 79)(21 56 45 78)(22 55 46 77)(23 54 47 76)(24 53 48 75)
(1 52)(2 93)(3 66)(4 83)(5 56)(6 73)(7 70)(8 87)(9 60)(10 77)(11 50)(12 91)(13 64)(14 81)(15 54)(16 95)(17 68)(18 85)(19 58)(20 75)(21 72)(22 89)(23 62)(24 79)(25 74)(26 71)(27 88)(28 61)(29 78)(30 51)(31 92)(32 65)(33 82)(34 55)(35 96)(36 69)(37 86)(38 59)(39 76)(40 49)(41 90)(42 63)(43 80)(44 53)(45 94)(46 67)(47 84)(48 57)
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), (1,52,25,74)(2,51,26,73)(3,50,27,96)(4,49,28,95)(5,72,29,94)(6,71,30,93)(7,70,31,92)(8,69,32,91)(9,68,33,90)(10,67,34,89)(11,66,35,88)(12,65,36,87)(13,64,37,86)(14,63,38,85)(15,62,39,84)(16,61,40,83)(17,60,41,82)(18,59,42,81)(19,58,43,80)(20,57,44,79)(21,56,45,78)(22,55,46,77)(23,54,47,76)(24,53,48,75), (1,52)(2,93)(3,66)(4,83)(5,56)(6,73)(7,70)(8,87)(9,60)(10,77)(11,50)(12,91)(13,64)(14,81)(15,54)(16,95)(17,68)(18,85)(19,58)(20,75)(21,72)(22,89)(23,62)(24,79)(25,74)(26,71)(27,88)(28,61)(29,78)(30,51)(31,92)(32,65)(33,82)(34,55)(35,96)(36,69)(37,86)(38,59)(39,76)(40,49)(41,90)(42,63)(43,80)(44,53)(45,94)(46,67)(47,84)(48,57)>;
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), (1,52,25,74)(2,51,26,73)(3,50,27,96)(4,49,28,95)(5,72,29,94)(6,71,30,93)(7,70,31,92)(8,69,32,91)(9,68,33,90)(10,67,34,89)(11,66,35,88)(12,65,36,87)(13,64,37,86)(14,63,38,85)(15,62,39,84)(16,61,40,83)(17,60,41,82)(18,59,42,81)(19,58,43,80)(20,57,44,79)(21,56,45,78)(22,55,46,77)(23,54,47,76)(24,53,48,75), (1,52)(2,93)(3,66)(4,83)(5,56)(6,73)(7,70)(8,87)(9,60)(10,77)(11,50)(12,91)(13,64)(14,81)(15,54)(16,95)(17,68)(18,85)(19,58)(20,75)(21,72)(22,89)(23,62)(24,79)(25,74)(26,71)(27,88)(28,61)(29,78)(30,51)(31,92)(32,65)(33,82)(34,55)(35,96)(36,69)(37,86)(38,59)(39,76)(40,49)(41,90)(42,63)(43,80)(44,53)(45,94)(46,67)(47,84)(48,57) );
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)], [(1,52,25,74),(2,51,26,73),(3,50,27,96),(4,49,28,95),(5,72,29,94),(6,71,30,93),(7,70,31,92),(8,69,32,91),(9,68,33,90),(10,67,34,89),(11,66,35,88),(12,65,36,87),(13,64,37,86),(14,63,38,85),(15,62,39,84),(16,61,40,83),(17,60,41,82),(18,59,42,81),(19,58,43,80),(20,57,44,79),(21,56,45,78),(22,55,46,77),(23,54,47,76),(24,53,48,75)], [(1,52),(2,93),(3,66),(4,83),(5,56),(6,73),(7,70),(8,87),(9,60),(10,77),(11,50),(12,91),(13,64),(14,81),(15,54),(16,95),(17,68),(18,85),(19,58),(20,75),(21,72),(22,89),(23,62),(24,79),(25,74),(26,71),(27,88),(28,61),(29,78),(30,51),(31,92),(32,65),(33,82),(34,55),(35,96),(36,69),(37,86),(38,59),(39,76),(40,49),(41,90),(42,63),(43,80),(44,53),(45,94),(46,67),(47,84),(48,57)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4oD4 | C4oD8 | C4oD12 | C8:C22 | D4:2S3 | S3xD4 | D8:3S3 | Q8:3D6 |
| kernel | C24:1C4:C2 | C12.Q8 | C24:1C4 | D6:C8 | D4:Dic3 | C3xD4:C4 | C4:C4:7S3 | D6:3D4 | D4:C4 | C2xDic3 | C22xS3 | C4:C4 | C2xC8 | C2xD4 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C24:1C4:C2 ►in GL4(F73) generated by
| 51 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 66 | 66 |
| 0 | 0 | 7 | 59 |
| 0 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 43 |
| 0 | 0 | 30 | 13 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 13 | 30 |
G:=sub<GL(4,GF(73))| [51,0,0,0,0,63,0,0,0,0,66,7,0,0,66,59],[0,1,0,0,72,0,0,0,0,0,60,30,0,0,43,13],[0,1,0,0,1,0,0,0,0,0,43,13,0,0,60,30] >;
C24:1C4:C2 in GAP, Magma, Sage, TeX
C_{24}\rtimes_1C_4\rtimes C_2 % in TeX
G:=Group("C24:1C4:C2"); // GroupNames label
G:=SmallGroup(192,343);
// by ID
G=gap.SmallGroup(192,343);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,926,219,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^19*b^2,c*b*c=b^-1>;
// generators/relations