metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12:2C8, C12.54D8, C12.49SD16, C42.192D6, C12.3M4(2), C4:C8:1S3, C3:2(D4:C8), C4.1(S3xC8), C6.10C4wrC2, C12.3(C2xC8), C2.7(D6:C8), (C4xD12).6C2, C4.1(C8:S3), C4.27(D4:S3), (C2xD12).11C4, (C2xC4).109D12, (C2xC12).225D4, C6.5(C22:C8), C4:Dic3.14C4, C6.3(D4:C4), (C4xC12).41C22, C2.1(C6.D8), C2.1(D12:C4), C22.34(D6:C4), C4.15(Q8:2S3), (C4xC3:C8):1C2, (C3xC4:C8):1C2, (C2xC4).65(C4xS3), (C2xC12).48(C2xC4), (C2xC4).265(C3:D4), (C2xC6).45(C22:C4), SmallGroup(192,42)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:2C8
G = < a,b,c | a12=b2=c8=1, bab=a-1, cac-1=a7, cbc-1=a3b >
Subgroups: 264 in 82 conjugacy classes, 35 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, C23, Dic3, C12, C12, D6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C22xC4, C2xD4, C3:C8, C24, C4xS3, D12, D12, C2xDic3, C2xC12, C22xS3, C4xC8, C4:C8, C4xD4, C2xC3:C8, C4:Dic3, D6:C4, C4xC12, C2xC24, S3xC2xC4, C2xD12, D4:C8, C4xC3:C8, C3xC4:C8, C4xD12, D12:2C8
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, D6, C22:C4, C2xC8, M4(2), D8, SD16, C4xS3, D12, C3:D4, C22:C8, D4:C4, C4wrC2, S3xC8, C8:S3, D6:C4, D4:S3, Q8:2S3, D4:C8, C6.D8, D6:C8, D12:C4, D12:2C8
►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 76)(2 75)(3 74)(4 73)(5 84)(6 83)(7 82)(8 81)(9 80)(10 79)(11 78)(12 77)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 60)(22 59)(23 58)(24 57)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(61 89)(62 88)(63 87)(64 86)(65 85)(66 96)(67 95)(68 94)(69 93)(70 92)(71 91)(72 90)
(1 69 58 33 77 85 24 38)(2 64 59 28 78 92 13 45)(3 71 60 35 79 87 14 40)(4 66 49 30 80 94 15 47)(5 61 50 25 81 89 16 42)(6 68 51 32 82 96 17 37)(7 63 52 27 83 91 18 44)(8 70 53 34 84 86 19 39)(9 65 54 29 73 93 20 46)(10 72 55 36 74 88 21 41)(11 67 56 31 75 95 22 48)(12 62 57 26 76 90 23 43)
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), (1,76)(2,75)(3,74)(4,73)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,60)(22,59)(23,58)(24,57)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(61,89)(62,88)(63,87)(64,86)(65,85)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90), (1,69,58,33,77,85,24,38)(2,64,59,28,78,92,13,45)(3,71,60,35,79,87,14,40)(4,66,49,30,80,94,15,47)(5,61,50,25,81,89,16,42)(6,68,51,32,82,96,17,37)(7,63,52,27,83,91,18,44)(8,70,53,34,84,86,19,39)(9,65,54,29,73,93,20,46)(10,72,55,36,74,88,21,41)(11,67,56,31,75,95,22,48)(12,62,57,26,76,90,23,43)>;
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), (1,76)(2,75)(3,74)(4,73)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,60)(22,59)(23,58)(24,57)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(61,89)(62,88)(63,87)(64,86)(65,85)(66,96)(67,95)(68,94)(69,93)(70,92)(71,91)(72,90), (1,69,58,33,77,85,24,38)(2,64,59,28,78,92,13,45)(3,71,60,35,79,87,14,40)(4,66,49,30,80,94,15,47)(5,61,50,25,81,89,16,42)(6,68,51,32,82,96,17,37)(7,63,52,27,83,91,18,44)(8,70,53,34,84,86,19,39)(9,65,54,29,73,93,20,46)(10,72,55,36,74,88,21,41)(11,67,56,31,75,95,22,48)(12,62,57,26,76,90,23,43) );
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)], [(1,76),(2,75),(3,74),(4,73),(5,84),(6,83),(7,82),(8,81),(9,80),(10,79),(11,78),(12,77),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,60),(22,59),(23,58),(24,57),(25,42),(26,41),(27,40),(28,39),(29,38),(30,37),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(61,89),(62,88),(63,87),(64,86),(65,85),(66,96),(67,95),(68,94),(69,93),(70,92),(71,91),(72,90)], [(1,69,58,33,77,85,24,38),(2,64,59,28,78,92,13,45),(3,71,60,35,79,87,14,40),(4,66,49,30,80,94,15,47),(5,61,50,25,81,89,16,42),(6,68,51,32,82,96,17,37),(7,63,52,27,83,91,18,44),(8,70,53,34,84,86,19,39),(9,65,54,29,73,93,20,46),(10,72,55,36,74,88,21,41),(11,67,56,31,75,95,22,48),(12,62,57,26,76,90,23,43)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D8 | SD16 | C4xS3 | D12 | C3:D4 | C4wrC2 | S3xC8 | C8:S3 | D4:S3 | Q8:2S3 | D12:C4 |
| kernel | D12:2C8 | C4xC3:C8 | C3xC4:C8 | C4xD12 | C4:Dic3 | C2xD12 | D12 | C4:C8 | C2xC12 | C42 | C12 | C12 | C12 | C2xC4 | C2xC4 | C2xC4 | C6 | C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of D12:2C8 ►in GL4(F73) generated by
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 67 | 67 |
| 0 | 0 | 67 | 6 |
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,1,1,0,0,0,0,0,1,0,0,1,0],[63,0,0,0,0,63,0,0,0,0,67,67,0,0,67,6] >;
D12:2C8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_2C_8 % in TeX
G:=Group("D12:2C8"); // GroupNames label
G:=SmallGroup(192,42);
// by ID
G=gap.SmallGroup(192,42);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,100,1123,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b>;
// generators/relations