metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12:3D4, Dic3:2D8, (C2xD24):5C2, C3:1(C4:D8), C4.88(S3xD4), C2.11(S3xD8), C6.25(C2xD8), (C2xC8).12D6, D4:C4:8S3, C12:3D4:2C2, Dic3:C8:7C2, C4:C4.140D6, (C2xD4).31D6, C6.D8:8C2, Dic3:5D4:3C2, C4.3(C4oD12), C12.112(C2xD4), C12.11(C4oD4), C6.19(C4:D4), C2.13(Q8:3D6), C6.58(C8:C22), (C2xC24).12C22, (C2xDic3).24D4, (C6xD4).47C22, C22.183(S3xD4), C2.22(Dic3:D4), (C2xC12).226C23, (C2xD12).54C22, (C4xDic3).13C22, (C2xD4:S3):6C2, (C3xD4:C4):8C2, (C2xC6).239(C2xD4), (C2xC3:C8).24C22, (C3xC4:C4).27C22, (C2xC4).333(C22xS3), SmallGroup(192,345)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:3D4
G = < a,b,c,d | a12=b2=c4=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a4b, dbd=a7b, dcd=c-1 >
Subgroups: 536 in 140 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, C23, Dic3, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xC8, D8, C22xC4, C2xD4, C2xD4, C3:C8, C24, C4xS3, D12, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, D4:C4, D4:C4, C4:C8, C4xD4, C4:1D4, C2xD8, D24, C2xC3:C8, C4xDic3, D6:C4, D4:S3, C3xC4:C4, C2xC24, S3xC2xC4, C2xD12, C2xC3:D4, C6xD4, C4:D8, C6.D8, Dic3:C8, C3xD4:C4, Dic3:5D4, C2xD24, C2xD4:S3, C12:3D4, D12:3D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C4oD4, C22xS3, C4:D4, C2xD8, C8:C22, C4oD12, S3xD4, C4:D8, Dic3:D4, S3xD8, Q8:3D6, D12:3D4
►On 96 points
Generators in S96
(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 66)(2 65)(3 64)(4 63)(5 62)(6 61)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 87)(14 86)(15 85)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 90)(23 89)(24 88)(25 75)(26 74)(27 73)(28 84)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 52)(38 51)(39 50)(40 49)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)
(1 80 19 53)(2 73 20 58)(3 78 21 51)(4 83 22 56)(5 76 23 49)(6 81 24 54)(7 74 13 59)(8 79 14 52)(9 84 15 57)(10 77 16 50)(11 82 17 55)(12 75 18 60)(25 90 41 63)(26 95 42 68)(27 88 43 61)(28 93 44 66)(29 86 45 71)(30 91 46 64)(31 96 47 69)(32 89 48 62)(33 94 37 67)(34 87 38 72)(35 92 39 65)(36 85 40 70)
(2 12)(3 11)(4 10)(5 9)(6 8)(14 24)(15 23)(16 22)(17 21)(18 20)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(49 84)(50 83)(51 82)(52 81)(53 80)(54 79)(55 78)(56 77)(57 76)(58 75)(59 74)(60 73)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)(85 94)(86 93)(87 92)(88 91)(89 90)(95 96)
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), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,87)(14,86)(15,85)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,90)(23,89)(24,88)(25,75)(26,74)(27,73)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,52)(38,51)(39,50)(40,49)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53), (1,80,19,53)(2,73,20,58)(3,78,21,51)(4,83,22,56)(5,76,23,49)(6,81,24,54)(7,74,13,59)(8,79,14,52)(9,84,15,57)(10,77,16,50)(11,82,17,55)(12,75,18,60)(25,90,41,63)(26,95,42,68)(27,88,43,61)(28,93,44,66)(29,86,45,71)(30,91,46,64)(31,96,47,69)(32,89,48,62)(33,94,37,67)(34,87,38,72)(35,92,39,65)(36,85,40,70), (2,12)(3,11)(4,10)(5,9)(6,8)(14,24)(15,23)(16,22)(17,21)(18,20)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)>;
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), (1,66)(2,65)(3,64)(4,63)(5,62)(6,61)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,87)(14,86)(15,85)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,90)(23,89)(24,88)(25,75)(26,74)(27,73)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,52)(38,51)(39,50)(40,49)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53), (1,80,19,53)(2,73,20,58)(3,78,21,51)(4,83,22,56)(5,76,23,49)(6,81,24,54)(7,74,13,59)(8,79,14,52)(9,84,15,57)(10,77,16,50)(11,82,17,55)(12,75,18,60)(25,90,41,63)(26,95,42,68)(27,88,43,61)(28,93,44,66)(29,86,45,71)(30,91,46,64)(31,96,47,69)(32,89,48,62)(33,94,37,67)(34,87,38,72)(35,92,39,65)(36,85,40,70), (2,12)(3,11)(4,10)(5,9)(6,8)(14,24)(15,23)(16,22)(17,21)(18,20)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,64)(62,63)(65,72)(66,71)(67,70)(68,69)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96) );
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)], [(1,66),(2,65),(3,64),(4,63),(5,62),(6,61),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,87),(14,86),(15,85),(16,96),(17,95),(18,94),(19,93),(20,92),(21,91),(22,90),(23,89),(24,88),(25,75),(26,74),(27,73),(28,84),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(37,52),(38,51),(39,50),(40,49),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53)], [(1,80,19,53),(2,73,20,58),(3,78,21,51),(4,83,22,56),(5,76,23,49),(6,81,24,54),(7,74,13,59),(8,79,14,52),(9,84,15,57),(10,77,16,50),(11,82,17,55),(12,75,18,60),(25,90,41,63),(26,95,42,68),(27,88,43,61),(28,93,44,66),(29,86,45,71),(30,91,46,64),(31,96,47,69),(32,89,48,62),(33,94,37,67),(34,87,38,72),(35,92,39,65),(36,85,40,70)], [(2,12),(3,11),(4,10),(5,9),(6,8),(14,24),(15,23),(16,22),(17,21),(18,20),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(49,84),(50,83),(51,82),(52,81),(53,80),(54,79),(55,78),(56,77),(57,76),(58,75),(59,74),(60,73),(61,64),(62,63),(65,72),(66,71),(67,70),(68,69),(85,94),(86,93),(87,92),(88,91),(89,90),(95,96)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 12 | 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 | D8 | C4oD4 | C4oD12 | C8:C22 | S3xD4 | S3xD4 | S3xD8 | Q8:3D6 |
| kernel | D12:3D4 | C6.D8 | Dic3:C8 | C3xD4:C4 | Dic3:5D4 | C2xD24 | C2xD4:S3 | C12:3D4 | D4:C4 | D12 | C2xDic3 | C4:C4 | C2xC8 | C2xD4 | Dic3 | C12 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D12:3D4 ►in GL6(F73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 0 | 0 | 0 | 16 | 57 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,16,0,0,0,0,16,57],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
D12:3D4 in GAP, Magma, Sage, TeX
D_{12}\rtimes_3D_4 % in TeX
G:=Group("D12:3D4"); // GroupNames label
G:=SmallGroup(192,345);
// by ID
G=gap.SmallGroup(192,345);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,219,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^4*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations