metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6:19D4, C6.692- 1+4, C4:D4:6S3, C4:C4.176D6, C3:3(Q8:5D4), C4.108(S3xD4), C22:C4.5D6, C4.D12:19C2, (D4xDic3):15C2, (C2xD4).151D6, C12.224(C2xD4), C6.61(C22xD4), C23.14D6:9C2, (C2xC6).142C24, D6:C4.11C22, C2.27(Q8oD12), Dic3.20(C2xD4), (C22xC4).234D6, Dic6:C4:19C2, C23.12D6:14C2, C22:2(D4:2S3), (C2xC12).500C23, (C22xDic6):16C2, (C6xD4).116C22, (C22xC6).13C23, C23.19(C22xS3), Dic3.D4:16C2, C23.11D6:17C2, Dic3:C4.13C22, (C22xS3).61C23, C4:Dic3.204C22, C22.163(S3xC23), (C2xDic3).65C23, (C4xDic3).89C22, (C22xC12).236C22, (C2xDic6).245C22, C6.D4.20C22, (C22xDic3).103C22, C2.34(C2xS3xD4), (C2xC6):4(C4oD4), (C3xC4:D4):7C2, (C4xC3:D4):14C2, C6.80(C2xC4oD4), (S3xC2xC4).81C22, (C2xD4:2S3):10C2, C2.31(C2xD4:2S3), (C3xC4:C4).138C22, (C2xC4).173(C22xS3), (C3xC22:C4).7C22, (C2xC3:D4).118C22, SmallGroup(192,1157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6:19D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a7, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 704 in 290 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, Dic6, Dic6, C4xS3, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C4xD4, C4xQ8, C4:D4, C4:D4, C22:Q8, C4.4D4, C22xQ8, C2xC4oD4, C4xDic3, C4xDic3, Dic3:C4, Dic3:C4, C4:Dic3, D6:C4, D6:C4, C6.D4, C6.D4, C3xC22:C4, C3xC4:C4, C2xDic6, C2xDic6, C2xDic6, S3xC2xC4, D4:2S3, C22xDic3, C2xC3:D4, C2xC3:D4, C22xC12, C6xD4, C6xD4, Q8:5D4, Dic3.D4, C23.11D6, Dic6:C4, C4.D12, C4xC3:D4, D4xDic3, C23.12D6, C23.14D6, C3xC4:D4, C22xDic6, C2xD4:2S3, Dic6:19D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2- 1+4, S3xD4, D4:2S3, S3xC23, Q8:5D4, C2xS3xD4, C2xD4:2S3, Q8oD12, Dic6:19D4
►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 70 7 64)(2 69 8 63)(3 68 9 62)(4 67 10 61)(5 66 11 72)(6 65 12 71)(13 26 19 32)(14 25 20 31)(15 36 21 30)(16 35 22 29)(17 34 23 28)(18 33 24 27)(37 77 43 83)(38 76 44 82)(39 75 45 81)(40 74 46 80)(41 73 47 79)(42 84 48 78)(49 96 55 90)(50 95 56 89)(51 94 57 88)(52 93 58 87)(53 92 59 86)(54 91 60 85)
(1 53 83 24)(2 60 84 19)(3 55 73 14)(4 50 74 21)(5 57 75 16)(6 52 76 23)(7 59 77 18)(8 54 78 13)(9 49 79 20)(10 56 80 15)(11 51 81 22)(12 58 82 17)(25 68 90 47)(26 63 91 42)(27 70 92 37)(28 65 93 44)(29 72 94 39)(30 67 95 46)(31 62 96 41)(32 69 85 48)(33 64 86 43)(34 71 87 38)(35 66 88 45)(36 61 89 40)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 25)(9 26)(10 27)(11 28)(12 29)(13 68)(14 69)(15 70)(16 71)(17 72)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(37 56)(38 57)(39 58)(40 59)(41 60)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 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), (1,70,7,64)(2,69,8,63)(3,68,9,62)(4,67,10,61)(5,66,11,72)(6,65,12,71)(13,26,19,32)(14,25,20,31)(15,36,21,30)(16,35,22,29)(17,34,23,28)(18,33,24,27)(37,77,43,83)(38,76,44,82)(39,75,45,81)(40,74,46,80)(41,73,47,79)(42,84,48,78)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85), (1,53,83,24)(2,60,84,19)(3,55,73,14)(4,50,74,21)(5,57,75,16)(6,52,76,23)(7,59,77,18)(8,54,78,13)(9,49,79,20)(10,56,80,15)(11,51,81,22)(12,58,82,17)(25,68,90,47)(26,63,91,42)(27,70,92,37)(28,65,93,44)(29,72,94,39)(30,67,95,46)(31,62,96,41)(32,69,85,48)(33,64,86,43)(34,71,87,38)(35,66,88,45)(36,61,89,40), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,68)(14,69)(15,70)(16,71)(17,72)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(37,56)(38,57)(39,58)(40,59)(41,60)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96)>;
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), (1,70,7,64)(2,69,8,63)(3,68,9,62)(4,67,10,61)(5,66,11,72)(6,65,12,71)(13,26,19,32)(14,25,20,31)(15,36,21,30)(16,35,22,29)(17,34,23,28)(18,33,24,27)(37,77,43,83)(38,76,44,82)(39,75,45,81)(40,74,46,80)(41,73,47,79)(42,84,48,78)(49,96,55,90)(50,95,56,89)(51,94,57,88)(52,93,58,87)(53,92,59,86)(54,91,60,85), (1,53,83,24)(2,60,84,19)(3,55,73,14)(4,50,74,21)(5,57,75,16)(6,52,76,23)(7,59,77,18)(8,54,78,13)(9,49,79,20)(10,56,80,15)(11,51,81,22)(12,58,82,17)(25,68,90,47)(26,63,91,42)(27,70,92,37)(28,65,93,44)(29,72,94,39)(30,67,95,46)(31,62,96,41)(32,69,85,48)(33,64,86,43)(34,71,87,38)(35,66,88,45)(36,61,89,40), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,25)(9,26)(10,27)(11,28)(12,29)(13,68)(14,69)(15,70)(16,71)(17,72)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(37,56)(38,57)(39,58)(40,59)(41,60)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96) );
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)], [(1,70,7,64),(2,69,8,63),(3,68,9,62),(4,67,10,61),(5,66,11,72),(6,65,12,71),(13,26,19,32),(14,25,20,31),(15,36,21,30),(16,35,22,29),(17,34,23,28),(18,33,24,27),(37,77,43,83),(38,76,44,82),(39,75,45,81),(40,74,46,80),(41,73,47,79),(42,84,48,78),(49,96,55,90),(50,95,56,89),(51,94,57,88),(52,93,58,87),(53,92,59,86),(54,91,60,85)], [(1,53,83,24),(2,60,84,19),(3,55,73,14),(4,50,74,21),(5,57,75,16),(6,52,76,23),(7,59,77,18),(8,54,78,13),(9,49,79,20),(10,56,80,15),(11,51,81,22),(12,58,82,17),(25,68,90,47),(26,63,91,42),(27,70,92,37),(28,65,93,44),(29,72,94,39),(30,67,95,46),(31,62,96,41),(32,69,85,48),(33,64,86,43),(34,71,87,38),(35,66,88,45),(36,61,89,40)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,25),(9,26),(10,27),(11,28),(12,29),(13,68),(14,69),(15,70),(16,71),(17,72),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(37,56),(38,57),(39,58),(40,59),(41,60),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | C4oD4 | 2- 1+4 | S3xD4 | D4:2S3 | Q8oD12 |
| kernel | Dic6:19D4 | Dic3.D4 | C23.11D6 | Dic6:C4 | C4.D12 | C4xC3:D4 | D4xDic3 | C23.12D6 | C23.14D6 | C3xC4:D4 | C22xDic6 | C2xD4:2S3 | C4:D4 | Dic6 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | C2xC6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 3 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6:19D4 ►in GL6(F13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 5 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 10 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,10,8,0,0,0,0,0,0,0,12,0,0,0,0,12,0],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,2,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;
Dic6:19D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_{19}D_4 % in TeX
G:=Group("Dic6:19D4"); // GroupNames label
G:=SmallGroup(192,1157);
// by ID
G=gap.SmallGroup(192,1157);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,477,232,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations