metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.12D4, C4.97(S3xD4), (C2xC8).20D6, Dic3:C8:8C2, C4:C4.157D6, Q8:C4:8S3, (C2xD24).3C2, (C2xQ8).48D6, Dic3:5D4:5C2, C6.72(C4oD8), C4.8(C4oD12), C12.129(C2xD4), C6.D8:13C2, C3:2(D4.2D4), C12.24(C4oD4), C12.23D4:2C2, C6.28(C4:D4), C2.19(Q8:3D6), C6.66(C8:C22), (C2xC24).20C22, (C2xDic3).36D4, C22.208(S3xD4), (C6xQ8).42C22, C2.31(Dic3:D4), (C2xC12).259C23, (C2xD12).69C22, C2.11(D24:C2), (C4xDic3).26C22, (C3xQ8:C4):8C2, (C2xQ8:2S3):6C2, (C2xC6).272(C2xD4), (C2xC3:C8).49C22, (C3xC4:C4).60C22, (C2xC4).366(C22xS3), SmallGroup(192,378)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.12D4
G = < a,b,c,d | a12=b2=c4=1, d2=a3, bab=a-1, cac-1=a7, ad=da, bc=cb, dbd-1=a3b, dcd-1=a9c-1 >
Subgroups: 440 in 124 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, D6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xC8, D8, SD16, C22xC4, C2xD4, C2xQ8, C3:C8, C24, C4xS3, D12, D12, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, D4:C4, Q8:C4, C4:C8, C4xD4, C4.4D4, C2xD8, C2xSD16, D24, C2xC3:C8, C4xDic3, D6:C4, Q8:2S3, C3xC4:C4, C2xC24, S3xC2xC4, C2xD12, C6xQ8, D4.2D4, C6.D8, Dic3:C8, C3xQ8:C4, Dic3:5D4, C2xD24, C2xQ8:2S3, C12.23D4, D12.12D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C22xS3, C4:D4, C4oD8, C8:C22, C4oD12, S3xD4, D4.2D4, Dic3:D4, Q8:3D6, D24:C2, D12.12D4
►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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 24)(9 23)(10 22)(11 21)(12 20)(25 40)(26 39)(27 38)(28 37)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 84)(59 83)(60 82)(61 93)(62 92)(63 91)(64 90)(65 89)(66 88)(67 87)(68 86)(69 85)(70 96)(71 95)(72 94)
(1 37 93 51)(2 44 94 58)(3 39 95 53)(4 46 96 60)(5 41 85 55)(6 48 86 50)(7 43 87 57)(8 38 88 52)(9 45 89 59)(10 40 90 54)(11 47 91 49)(12 42 92 56)(13 34 67 73)(14 29 68 80)(15 36 69 75)(16 31 70 82)(17 26 71 77)(18 33 72 84)(19 28 61 79)(20 35 62 74)(21 30 63 81)(22 25 64 76)(23 32 65 83)(24 27 66 78)
(1 68 4 71 7 62 10 65)(2 69 5 72 8 63 11 66)(3 70 6 61 9 64 12 67)(13 95 16 86 19 89 22 92)(14 96 17 87 20 90 23 93)(15 85 18 88 21 91 24 94)(25 39 28 42 31 45 34 48)(26 40 29 43 32 46 35 37)(27 41 30 44 33 47 36 38)(49 75 52 78 55 81 58 84)(50 76 53 79 56 82 59 73)(51 77 54 80 57 83 60 74)
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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,24)(9,23)(10,22)(11,21)(12,20)(25,40)(26,39)(27,38)(28,37)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,84)(59,83)(60,82)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94), (1,37,93,51)(2,44,94,58)(3,39,95,53)(4,46,96,60)(5,41,85,55)(6,48,86,50)(7,43,87,57)(8,38,88,52)(9,45,89,59)(10,40,90,54)(11,47,91,49)(12,42,92,56)(13,34,67,73)(14,29,68,80)(15,36,69,75)(16,31,70,82)(17,26,71,77)(18,33,72,84)(19,28,61,79)(20,35,62,74)(21,30,63,81)(22,25,64,76)(23,32,65,83)(24,27,66,78), (1,68,4,71,7,62,10,65)(2,69,5,72,8,63,11,66)(3,70,6,61,9,64,12,67)(13,95,16,86,19,89,22,92)(14,96,17,87,20,90,23,93)(15,85,18,88,21,91,24,94)(25,39,28,42,31,45,34,48)(26,40,29,43,32,46,35,37)(27,41,30,44,33,47,36,38)(49,75,52,78,55,81,58,84)(50,76,53,79,56,82,59,73)(51,77,54,80,57,83,60,74)>;
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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,24)(9,23)(10,22)(11,21)(12,20)(25,40)(26,39)(27,38)(28,37)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,84)(59,83)(60,82)(61,93)(62,92)(63,91)(64,90)(65,89)(66,88)(67,87)(68,86)(69,85)(70,96)(71,95)(72,94), (1,37,93,51)(2,44,94,58)(3,39,95,53)(4,46,96,60)(5,41,85,55)(6,48,86,50)(7,43,87,57)(8,38,88,52)(9,45,89,59)(10,40,90,54)(11,47,91,49)(12,42,92,56)(13,34,67,73)(14,29,68,80)(15,36,69,75)(16,31,70,82)(17,26,71,77)(18,33,72,84)(19,28,61,79)(20,35,62,74)(21,30,63,81)(22,25,64,76)(23,32,65,83)(24,27,66,78), (1,68,4,71,7,62,10,65)(2,69,5,72,8,63,11,66)(3,70,6,61,9,64,12,67)(13,95,16,86,19,89,22,92)(14,96,17,87,20,90,23,93)(15,85,18,88,21,91,24,94)(25,39,28,42,31,45,34,48)(26,40,29,43,32,46,35,37)(27,41,30,44,33,47,36,38)(49,75,52,78,55,81,58,84)(50,76,53,79,56,82,59,73)(51,77,54,80,57,83,60,74) );
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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,24),(9,23),(10,22),(11,21),(12,20),(25,40),(26,39),(27,38),(28,37),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,84),(59,83),(60,82),(61,93),(62,92),(63,91),(64,90),(65,89),(66,88),(67,87),(68,86),(69,85),(70,96),(71,95),(72,94)], [(1,37,93,51),(2,44,94,58),(3,39,95,53),(4,46,96,60),(5,41,85,55),(6,48,86,50),(7,43,87,57),(8,38,88,52),(9,45,89,59),(10,40,90,54),(11,47,91,49),(12,42,92,56),(13,34,67,73),(14,29,68,80),(15,36,69,75),(16,31,70,82),(17,26,71,77),(18,33,72,84),(19,28,61,79),(20,35,62,74),(21,30,63,81),(22,25,64,76),(23,32,65,83),(24,27,66,78)], [(1,68,4,71,7,62,10,65),(2,69,5,72,8,63,11,66),(3,70,6,61,9,64,12,67),(13,95,16,86,19,89,22,92),(14,96,17,87,20,90,23,93),(15,85,18,88,21,91,24,94),(25,39,28,42,31,45,34,48),(26,40,29,43,32,46,35,37),(27,41,30,44,33,47,36,38),(49,75,52,78,55,81,58,84),(50,76,53,79,56,82,59,73),(51,77,54,80,57,83,60,74)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 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 | S3xD4 | S3xD4 | Q8:3D6 | D24:C2 |
| kernel | D12.12D4 | C6.D8 | Dic3:C8 | C3xQ8:C4 | Dic3:5D4 | C2xD24 | C2xQ8:2S3 | C12.23D4 | Q8:C4 | D12 | C2xDic3 | C4:C4 | C2xC8 | C2xQ8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12.12D4 ►in GL4(F73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 2 |
| 0 | 0 | 72 | 1 |
| 66 | 66 | 0 | 0 |
| 59 | 7 | 0 | 0 |
| 0 | 0 | 41 | 32 |
| 0 | 0 | 57 | 32 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 61 | 12 |
| 0 | 0 | 67 | 12 |
| 66 | 14 | 0 | 0 |
| 59 | 7 | 0 | 0 |
| 0 | 0 | 32 | 41 |
| 0 | 0 | 16 | 0 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,72,72,0,0,2,1],[66,59,0,0,66,7,0,0,0,0,41,57,0,0,32,32],[46,0,0,0,0,46,0,0,0,0,61,67,0,0,12,12],[66,59,0,0,14,7,0,0,0,0,32,16,0,0,41,0] >;
D12.12D4 in GAP, Magma, Sage, TeX
D_{12}._{12}D_4 % in TeX
G:=Group("D12.12D4"); // GroupNames label
G:=SmallGroup(192,378);
// by ID
G=gap.SmallGroup(192,378);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,184,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^-1>;
// generators/relations