metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6:9D4, C12:3SD16, C42.75D6, C4.54(S3xD4), C4:2(D4.S3), (C2xD4).58D6, C12:C8:32C2, C3:4(C4:SD16), C4:1D4.6S3, C12.33(C2xD4), (C4xDic6):22C2, (C2xC12).149D4, C6.58(C2xSD16), C12.77(C4oD4), C4.4(D4:2S3), D4:Dic3:23C2, C6.96(C8:C22), (C6xD4).74C22, C2.13(D6:3D4), C6.104(C4:D4), (C2xC12).393C23, (C4xC12).123C22, C2.17(D12:6C22), C4:Dic3.345C22, (C2xDic6).274C22, (C2xD4.S3):14C2, (C3xC4:1D4).4C2, (C2xC6).524(C2xD4), C2.12(C2xD4.S3), (C2xC3:C8).132C22, (C2xC4).186(C3:D4), (C2xC4).491(C22xS3), C22.197(C2xC3:D4), SmallGroup(192,634)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6:9D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a9b, dcd=c-1 >
Subgroups: 352 in 128 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, C12, C12, C2xC6, C2xC6, C42, C42, C4:C4, C2xC8, SD16, C2xD4, C2xD4, C2xQ8, C3:C8, Dic6, Dic6, C2xDic3, C2xC12, C3xD4, C22xC6, D4:C4, C4:C8, C4xQ8, C4:1D4, C2xSD16, C2xC3:C8, C4xDic3, Dic3:C4, C4:Dic3, D4.S3, C4xC12, C2xDic6, C6xD4, C6xD4, C4:SD16, C12:C8, D4:Dic3, C4xDic6, C2xD4.S3, C3xC4:1D4, Dic6:9D4
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, C2xSD16, C8:C22, D4.S3, S3xD4, D4:2S3, C2xC3:D4, C4:SD16, D12:6C22, C2xD4.S3, D6:3D4, Dic6:9D4
►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 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 71 19 65)(14 70 20 64)(15 69 21 63)(16 68 22 62)(17 67 23 61)(18 66 24 72)(37 80 43 74)(38 79 44 73)(39 78 45 84)(40 77 46 83)(41 76 47 82)(42 75 48 81)(49 90 55 96)(50 89 56 95)(51 88 57 94)(52 87 58 93)(53 86 59 92)(54 85 60 91)
(1 13 52 82)(2 14 53 83)(3 15 54 84)(4 16 55 73)(5 17 56 74)(6 18 57 75)(7 19 58 76)(8 20 59 77)(9 21 60 78)(10 22 49 79)(11 23 50 80)(12 24 51 81)(25 63 91 45)(26 64 92 46)(27 65 93 47)(28 66 94 48)(29 67 95 37)(30 68 96 38)(31 69 85 39)(32 70 86 40)(33 71 87 41)(34 72 88 42)(35 61 89 43)(36 62 90 44)
(2 8)(4 10)(6 12)(13 82)(14 77)(15 84)(16 79)(17 74)(18 81)(19 76)(20 83)(21 78)(22 73)(23 80)(24 75)(25 28)(26 35)(27 30)(29 32)(31 34)(33 36)(37 70)(38 65)(39 72)(40 67)(41 62)(42 69)(43 64)(44 71)(45 66)(46 61)(47 68)(48 63)(49 55)(51 57)(53 59)(85 88)(86 95)(87 90)(89 92)(91 94)(93 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), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,71,19,65)(14,70,20,64)(15,69,21,63)(16,68,22,62)(17,67,23,61)(18,66,24,72)(37,80,43,74)(38,79,44,73)(39,78,45,84)(40,77,46,83)(41,76,47,82)(42,75,48,81)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91), (1,13,52,82)(2,14,53,83)(3,15,54,84)(4,16,55,73)(5,17,56,74)(6,18,57,75)(7,19,58,76)(8,20,59,77)(9,21,60,78)(10,22,49,79)(11,23,50,80)(12,24,51,81)(25,63,91,45)(26,64,92,46)(27,65,93,47)(28,66,94,48)(29,67,95,37)(30,68,96,38)(31,69,85,39)(32,70,86,40)(33,71,87,41)(34,72,88,42)(35,61,89,43)(36,62,90,44), (2,8)(4,10)(6,12)(13,82)(14,77)(15,84)(16,79)(17,74)(18,81)(19,76)(20,83)(21,78)(22,73)(23,80)(24,75)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,70)(38,65)(39,72)(40,67)(41,62)(42,69)(43,64)(44,71)(45,66)(46,61)(47,68)(48,63)(49,55)(51,57)(53,59)(85,88)(86,95)(87,90)(89,92)(91,94)(93,96)>;
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), (1,33,7,27)(2,32,8,26)(3,31,9,25)(4,30,10,36)(5,29,11,35)(6,28,12,34)(13,71,19,65)(14,70,20,64)(15,69,21,63)(16,68,22,62)(17,67,23,61)(18,66,24,72)(37,80,43,74)(38,79,44,73)(39,78,45,84)(40,77,46,83)(41,76,47,82)(42,75,48,81)(49,90,55,96)(50,89,56,95)(51,88,57,94)(52,87,58,93)(53,86,59,92)(54,85,60,91), (1,13,52,82)(2,14,53,83)(3,15,54,84)(4,16,55,73)(5,17,56,74)(6,18,57,75)(7,19,58,76)(8,20,59,77)(9,21,60,78)(10,22,49,79)(11,23,50,80)(12,24,51,81)(25,63,91,45)(26,64,92,46)(27,65,93,47)(28,66,94,48)(29,67,95,37)(30,68,96,38)(31,69,85,39)(32,70,86,40)(33,71,87,41)(34,72,88,42)(35,61,89,43)(36,62,90,44), (2,8)(4,10)(6,12)(13,82)(14,77)(15,84)(16,79)(17,74)(18,81)(19,76)(20,83)(21,78)(22,73)(23,80)(24,75)(25,28)(26,35)(27,30)(29,32)(31,34)(33,36)(37,70)(38,65)(39,72)(40,67)(41,62)(42,69)(43,64)(44,71)(45,66)(46,61)(47,68)(48,63)(49,55)(51,57)(53,59)(85,88)(86,95)(87,90)(89,92)(91,94)(93,96) );
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)], [(1,33,7,27),(2,32,8,26),(3,31,9,25),(4,30,10,36),(5,29,11,35),(6,28,12,34),(13,71,19,65),(14,70,20,64),(15,69,21,63),(16,68,22,62),(17,67,23,61),(18,66,24,72),(37,80,43,74),(38,79,44,73),(39,78,45,84),(40,77,46,83),(41,76,47,82),(42,75,48,81),(49,90,55,96),(50,89,56,95),(51,88,57,94),(52,87,58,93),(53,86,59,92),(54,85,60,91)], [(1,13,52,82),(2,14,53,83),(3,15,54,84),(4,16,55,73),(5,17,56,74),(6,18,57,75),(7,19,58,76),(8,20,59,77),(9,21,60,78),(10,22,49,79),(11,23,50,80),(12,24,51,81),(25,63,91,45),(26,64,92,46),(27,65,93,47),(28,66,94,48),(29,67,95,37),(30,68,96,38),(31,69,85,39),(32,70,86,40),(33,71,87,41),(34,72,88,42),(35,61,89,43),(36,62,90,44)], [(2,8),(4,10),(6,12),(13,82),(14,77),(15,84),(16,79),(17,74),(18,81),(19,76),(20,83),(21,78),(22,73),(23,80),(24,75),(25,28),(26,35),(27,30),(29,32),(31,34),(33,36),(37,70),(38,65),(39,72),(40,67),(41,62),(42,69),(43,64),(44,71),(45,66),(46,61),(47,68),(48,63),(49,55),(51,57),(53,59),(85,88),(86,95),(87,90),(89,92),(91,94),(93,96)]])
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 | 6F | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | C4oD4 | C3:D4 | C8:C22 | D4.S3 | S3xD4 | D4:2S3 | D12:6C22 |
| kernel | Dic6:9D4 | C12:C8 | D4:Dic3 | C4xDic6 | C2xD4.S3 | C3xC4:1D4 | C4:1D4 | Dic6 | C2xC12 | C42 | C2xD4 | C12 | C12 | C2xC4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 2 | 1 | 1 | 2 |
Matrix representation of Dic6:9D4 ►in GL6(F73)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 67 |
| 0 | 0 | 0 | 0 | 67 | 67 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 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 |
| 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 |
G:=sub<GL(6,GF(73))| [9,0,0,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,6,67,0,0,0,0,67,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
Dic6:9D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_9D_4 % in TeX
G:=Group("Dic6:9D4"); // GroupNames label
G:=SmallGroup(192,634);
// by ID
G=gap.SmallGroup(192,634);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations