metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4:5Dic6, C42.103D6, C6.132+ 1+4, (C3xD4):6Q8, C12:Q8:15C2, C4:C4.278D6, (C4xD4).11S3, C3:2(D4:3Q8), C12.42(C2xQ8), (C4xDic6):26C2, (C2xD4).242D6, (D4xC12).12C2, (C2xC6).83C24, C4.15(C2xDic6), C12.48D4:7C2, C6.13(C22xQ8), C22:C4.106D6, C4.Dic6:14C2, C12.6Q8:14C2, (D4xDic3).11C2, (C22xC4).218D6, C2.16(D4:6D6), C22.1(C2xDic6), (C2xC12).154C23, (C4xC12).146C22, Dic3.D4:7C2, (C6xD4).249C22, C4:Dic3.37C22, C2.15(C22xDic6), Dic3.19(C4oD4), C22.111(S3xC23), (C22xC6).153C23, (C22xC12).77C22, C23.173(C22xS3), (C2xDic3).33C23, (C2xDic6).25C22, (C4xDic3).72C22, C6.D4.8C22, Dic3:C4.108C22, (C22xDic3).91C22, (C2xC6).3(C2xQ8), C2.18(S3xC4oD4), C6.137(C2xC4oD4), (C2xDic3:C4):24C2, (C3xC4:C4).319C22, (C2xC4).154(C22xS3), (C3xC22:C4).104C22, SmallGroup(192,1098)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4:5Dic6
G = < a,b,c,d | a4=b2=c12=1, d2=c6, bab=cac-1=a-1, ad=da, cbc-1=dbd-1=a2b, dcd-1=c-1 >
Subgroups: 504 in 228 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2xC6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, Dic6, C2xDic3, C2xDic3, C2xDic3, C2xC12, C2xC12, C2xC12, C3xD4, C22xC6, C2xC4:C4, C4xD4, C4xD4, C4xQ8, C22:Q8, C42.C2, C4:Q8, C4xDic3, Dic3:C4, Dic3:C4, C4:Dic3, C4:Dic3, C6.D4, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, C2xDic6, C22xDic3, C22xC12, C6xD4, D4:3Q8, C4xDic6, C12.6Q8, Dic3.D4, C12:Q8, C4.Dic6, C2xDic3:C4, C12.48D4, D4xDic3, D4xC12, D4:5Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, C4oD4, C24, Dic6, C22xS3, C22xQ8, C2xC4oD4, 2+ 1+4, C2xDic6, S3xC23, D4:3Q8, C22xDic6, D4:6D6, S3xC4oD4, D4:5Dic6
►On 96 points
(1 57 31 44)(2 45 32 58)(3 59 33 46)(4 47 34 60)(5 49 35 48)(6 37 36 50)(7 51 25 38)(8 39 26 52)(9 53 27 40)(10 41 28 54)(11 55 29 42)(12 43 30 56)(13 80 94 64)(14 65 95 81)(15 82 96 66)(16 67 85 83)(17 84 86 68)(18 69 87 73)(19 74 88 70)(20 71 89 75)(21 76 90 72)(22 61 91 77)(23 78 92 62)(24 63 93 79)
(1 51)(2 39)(3 53)(4 41)(5 55)(6 43)(7 57)(8 45)(9 59)(10 47)(11 49)(12 37)(13 70)(14 75)(15 72)(16 77)(17 62)(18 79)(19 64)(20 81)(21 66)(22 83)(23 68)(24 73)(25 44)(26 58)(27 46)(28 60)(29 48)(30 50)(31 38)(32 52)(33 40)(34 54)(35 42)(36 56)(61 85)(63 87)(65 89)(67 91)(69 93)(71 95)(74 94)(76 96)(78 86)(80 88)(82 90)(84 92)
(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 13 7 19)(2 24 8 18)(3 23 9 17)(4 22 10 16)(5 21 11 15)(6 20 12 14)(25 88 31 94)(26 87 32 93)(27 86 33 92)(28 85 34 91)(29 96 35 90)(30 95 36 89)(37 71 43 65)(38 70 44 64)(39 69 45 63)(40 68 46 62)(41 67 47 61)(42 66 48 72)(49 76 55 82)(50 75 56 81)(51 74 57 80)(52 73 58 79)(53 84 59 78)(54 83 60 77)
G:=sub<Sym(96)| (1,57,31,44)(2,45,32,58)(3,59,33,46)(4,47,34,60)(5,49,35,48)(6,37,36,50)(7,51,25,38)(8,39,26,52)(9,53,27,40)(10,41,28,54)(11,55,29,42)(12,43,30,56)(13,80,94,64)(14,65,95,81)(15,82,96,66)(16,67,85,83)(17,84,86,68)(18,69,87,73)(19,74,88,70)(20,71,89,75)(21,76,90,72)(22,61,91,77)(23,78,92,62)(24,63,93,79), (1,51)(2,39)(3,53)(4,41)(5,55)(6,43)(7,57)(8,45)(9,59)(10,47)(11,49)(12,37)(13,70)(14,75)(15,72)(16,77)(17,62)(18,79)(19,64)(20,81)(21,66)(22,83)(23,68)(24,73)(25,44)(26,58)(27,46)(28,60)(29,48)(30,50)(31,38)(32,52)(33,40)(34,54)(35,42)(36,56)(61,85)(63,87)(65,89)(67,91)(69,93)(71,95)(74,94)(76,96)(78,86)(80,88)(82,90)(84,92), (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), (1,13,7,19)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,88,31,94)(26,87,32,93)(27,86,33,92)(28,85,34,91)(29,96,35,90)(30,95,36,89)(37,71,43,65)(38,70,44,64)(39,69,45,63)(40,68,46,62)(41,67,47,61)(42,66,48,72)(49,76,55,82)(50,75,56,81)(51,74,57,80)(52,73,58,79)(53,84,59,78)(54,83,60,77)>;
G:=Group( (1,57,31,44)(2,45,32,58)(3,59,33,46)(4,47,34,60)(5,49,35,48)(6,37,36,50)(7,51,25,38)(8,39,26,52)(9,53,27,40)(10,41,28,54)(11,55,29,42)(12,43,30,56)(13,80,94,64)(14,65,95,81)(15,82,96,66)(16,67,85,83)(17,84,86,68)(18,69,87,73)(19,74,88,70)(20,71,89,75)(21,76,90,72)(22,61,91,77)(23,78,92,62)(24,63,93,79), (1,51)(2,39)(3,53)(4,41)(5,55)(6,43)(7,57)(8,45)(9,59)(10,47)(11,49)(12,37)(13,70)(14,75)(15,72)(16,77)(17,62)(18,79)(19,64)(20,81)(21,66)(22,83)(23,68)(24,73)(25,44)(26,58)(27,46)(28,60)(29,48)(30,50)(31,38)(32,52)(33,40)(34,54)(35,42)(36,56)(61,85)(63,87)(65,89)(67,91)(69,93)(71,95)(74,94)(76,96)(78,86)(80,88)(82,90)(84,92), (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), (1,13,7,19)(2,24,8,18)(3,23,9,17)(4,22,10,16)(5,21,11,15)(6,20,12,14)(25,88,31,94)(26,87,32,93)(27,86,33,92)(28,85,34,91)(29,96,35,90)(30,95,36,89)(37,71,43,65)(38,70,44,64)(39,69,45,63)(40,68,46,62)(41,67,47,61)(42,66,48,72)(49,76,55,82)(50,75,56,81)(51,74,57,80)(52,73,58,79)(53,84,59,78)(54,83,60,77) );
G=PermutationGroup([[(1,57,31,44),(2,45,32,58),(3,59,33,46),(4,47,34,60),(5,49,35,48),(6,37,36,50),(7,51,25,38),(8,39,26,52),(9,53,27,40),(10,41,28,54),(11,55,29,42),(12,43,30,56),(13,80,94,64),(14,65,95,81),(15,82,96,66),(16,67,85,83),(17,84,86,68),(18,69,87,73),(19,74,88,70),(20,71,89,75),(21,76,90,72),(22,61,91,77),(23,78,92,62),(24,63,93,79)], [(1,51),(2,39),(3,53),(4,41),(5,55),(6,43),(7,57),(8,45),(9,59),(10,47),(11,49),(12,37),(13,70),(14,75),(15,72),(16,77),(17,62),(18,79),(19,64),(20,81),(21,66),(22,83),(23,68),(24,73),(25,44),(26,58),(27,46),(28,60),(29,48),(30,50),(31,38),(32,52),(33,40),(34,54),(35,42),(36,56),(61,85),(63,87),(65,89),(67,91),(69,93),(71,95),(74,94),(76,96),(78,86),(80,88),(82,90),(84,92)], [(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)], [(1,13,7,19),(2,24,8,18),(3,23,9,17),(4,22,10,16),(5,21,11,15),(6,20,12,14),(25,88,31,94),(26,87,32,93),(27,86,33,92),(28,85,34,91),(29,96,35,90),(30,95,36,89),(37,71,43,65),(38,70,44,64),(39,69,45,63),(40,68,46,62),(41,67,47,61),(42,66,48,72),(49,76,55,82),(50,75,56,81),(51,74,57,80),(52,73,58,79),(53,84,59,78),(54,83,60,77)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | ··· | 4Q | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | D6 | C4oD4 | Dic6 | 2+ 1+4 | D4:6D6 | S3xC4oD4 |
| kernel | D4:5Dic6 | C4xDic6 | C12.6Q8 | Dic3.D4 | C12:Q8 | C4.Dic6 | C2xDic3:C4 | C12.48D4 | D4xDic3 | D4xC12 | C4xD4 | C3xD4 | C42 | C22:C4 | C4:C4 | C22xC4 | C2xD4 | Dic3 | D4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of D4:5Dic6 ►in GL6(F13)
| 1 | 0 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 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 | 1 | 0 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,12,0,0,0,0,2,12,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[5,8,0,0,0,0,0,8,0,0,0,0,0,0,3,10,0,0,0,0,7,10,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
D4:5Dic6 in GAP, Magma, Sage, TeX
D_4\rtimes_5{\rm Dic}_6 % in TeX
G:=Group("D4:5Dic6"); // GroupNames label
G:=SmallGroup(192,1098);
// by ID
G=gap.SmallGroup(192,1098);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,387,675,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^12=1,d^2=c^6,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^-1>;
// generators/relations