metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12:7D8, D4:1D12, C42.49D6, (C3xD4):8D4, (C4xD4):3S3, C4:3(D4:S3), (D4xC12):3C2, C3:3(C4:D8), C6.52(C2xD8), C4:D12:8C2, C4:C4.243D6, C12:C8:23C2, (C2xC12).60D4, C4.13(C2xD12), C12.17(C2xD4), (C2xD4).190D6, C4.9(C4oD12), C6.D8:30C2, C12.50(C4oD4), C2.8(D4:D6), C6.64(C4:D4), (C4xC12).86C22, C2.12(C12:7D4), C6.109(C8:C22), (C2xC12).337C23, (C6xD4).232C22, (C2xD12).93C22, (C2xD4:S3):7C2, C2.7(C2xD4:S3), (C2xC6).468(C2xD4), (C2xC3:C8).93C22, (C2xC4).245(C3:D4), (C3xC4:C4).274C22, (C2xC4).437(C22xS3), C22.149(C2xC3:D4), SmallGroup(192,574)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12:7D8
G = < a,b,c | a12=b8=c2=1, bab-1=cac=a-1, cbc=b-1 >
Subgroups: 504 in 140 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, C23, C12, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, D8, C22xC4, C2xD4, C2xD4, C3:C8, D12, C2xC12, C2xC12, C3xD4, C3xD4, C22xS3, C22xC6, D4:C4, C4:C8, C4xD4, C4:1D4, C2xD8, C2xC3:C8, D4:S3, C4xC12, C3xC22:C4, C3xC4:C4, C2xD12, C2xD12, C22xC12, C6xD4, C4:D8, C12:C8, C6.D8, C4:D12, C2xD4:S3, D4xC12, C12:7D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C4oD4, D12, C3:D4, C22xS3, C4:D4, C2xD8, C8:C22, D4:S3, C2xD12, C4oD12, C2xC3:D4, C4:D8, C12:7D4, C2xD4:S3, D4:D6, C12:7D8
►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 69 89 22 38 30 81)(2 51 70 88 23 37 31 80)(3 50 71 87 24 48 32 79)(4 49 72 86 13 47 33 78)(5 60 61 85 14 46 34 77)(6 59 62 96 15 45 35 76)(7 58 63 95 16 44 36 75)(8 57 64 94 17 43 25 74)(9 56 65 93 18 42 26 73)(10 55 66 92 19 41 27 84)(11 54 67 91 20 40 28 83)(12 53 68 90 21 39 29 82)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 19)(14 18)(15 17)(20 24)(21 23)(25 62)(26 61)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 96)(44 95)(45 94)(46 93)(47 92)(48 91)(49 84)(50 83)(51 82)(52 81)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)
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), (1,52,69,89,22,38,30,81)(2,51,70,88,23,37,31,80)(3,50,71,87,24,48,32,79)(4,49,72,86,13,47,33,78)(5,60,61,85,14,46,34,77)(6,59,62,96,15,45,35,76)(7,58,63,95,16,44,36,75)(8,57,64,94,17,43,25,74)(9,56,65,93,18,42,26,73)(10,55,66,92,19,41,27,84)(11,54,67,91,20,40,28,83)(12,53,68,90,21,39,29,82), (2,12)(3,11)(4,10)(5,9)(6,8)(13,19)(14,18)(15,17)(20,24)(21,23)(25,62)(26,61)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)>;
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), (1,52,69,89,22,38,30,81)(2,51,70,88,23,37,31,80)(3,50,71,87,24,48,32,79)(4,49,72,86,13,47,33,78)(5,60,61,85,14,46,34,77)(6,59,62,96,15,45,35,76)(7,58,63,95,16,44,36,75)(8,57,64,94,17,43,25,74)(9,56,65,93,18,42,26,73)(10,55,66,92,19,41,27,84)(11,54,67,91,20,40,28,83)(12,53,68,90,21,39,29,82), (2,12)(3,11)(4,10)(5,9)(6,8)(13,19)(14,18)(15,17)(20,24)(21,23)(25,62)(26,61)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73) );
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)], [(1,52,69,89,22,38,30,81),(2,51,70,88,23,37,31,80),(3,50,71,87,24,48,32,79),(4,49,72,86,13,47,33,78),(5,60,61,85,14,46,34,77),(6,59,62,96,15,45,35,76),(7,58,63,95,16,44,36,75),(8,57,64,94,17,43,25,74),(9,56,65,93,18,42,26,73),(10,55,66,92,19,41,27,84),(11,54,67,91,20,40,28,83),(12,53,68,90,21,39,29,82)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,19),(14,18),(15,17),(20,24),(21,23),(25,62),(26,61),(27,72),(28,71),(29,70),(30,69),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,96),(44,95),(45,94),(46,93),(47,92),(48,91),(49,84),(50,83),(51,82),(52,81),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D8 | C4oD4 | C3:D4 | D12 | C4oD12 | C8:C22 | D4:S3 | D4:D6 |
| kernel | C12:7D8 | C12:C8 | C6.D8 | C4:D12 | C2xD4:S3 | D4xC12 | C4xD4 | C2xC12 | C3xD4 | C42 | C4:C4 | C2xD4 | C12 | C12 | C2xC4 | D4 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of C12:7D8 ►in GL6(F73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 19 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 57 | 16 | 0 | 0 | 0 | 0 |
| 57 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 2 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,19,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,72,72,0,0,0,0,2,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C12:7D8 in GAP, Magma, Sage, TeX
C_{12}\rtimes_7D_8 % in TeX
G:=Group("C12:7D8"); // GroupNames label
G:=SmallGroup(192,574);
// by ID
G=gap.SmallGroup(192,574);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations