metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3:8SD16, C8:8(C4xS3), C3:4(C4xSD16), C24:C2:5C4, C24:13(C2xC4), C6.49(C4xD4), C4.Q8:13S3, C4:C4.158D6, Dic6:6(C2xC4), (C8xDic3):8C2, D12.8(C2xC4), (C2xC8).255D6, C2.5(S3xSD16), C6.51(C4oD8), C6.34(C2xSD16), C6.D8.4C2, C22.81(S3xD4), Dic6:C4:6C2, Dic3:5D4.4C2, C12.25(C4oD4), C6.SD16:14C2, C12.40(C22xC4), C4.1(Q8:3S3), C2.9(Dic3:5D4), (C2xC12).268C23, (C2xC24).156C22, (C2xDic3).205D4, C2.5(Q8.7D6), (C2xD12).72C22, (C2xDic6).78C22, (C4xDic3).228C22, C4.40(S3xC2xC4), (C3xC4.Q8):6C2, (C2xC24:C2).9C2, (C2xC6).273(C2xD4), (C3xC4:C4).61C22, (C2xC3:C8).223C22, (C2xC4).371(C22xS3), SmallGroup(192,411)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3:8SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c3 >
Subgroups: 352 in 122 conjugacy classes, 51 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C3:C8, C24, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C4xC8, D4:C4, Q8:C4, C4.Q8, C4xD4, C4xQ8, C2xSD16, C24:C2, C2xC3:C8, C4xDic3, C4xDic3, Dic3:C4, D6:C4, C3xC4:C4, C2xC24, C2xDic6, S3xC2xC4, C2xD12, C4xSD16, C6.D8, C6.SD16, C8xDic3, C3xC4.Q8, Dic6:C4, Dic3:5D4, C2xC24:C2, Dic3:8SD16
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, SD16, C22xC4, C2xD4, C4oD4, C4xS3, C22xS3, C4xD4, C2xSD16, C4oD8, S3xC2xC4, S3xD4, Q8:3S3, C4xSD16, Dic3:5D4, S3xSD16, Q8.7D6, Dic3:8SD16
►On 96 points
(1 73 55 86 34 16)(2 74 56 87 35 9)(3 75 49 88 36 10)(4 76 50 81 37 11)(5 77 51 82 38 12)(6 78 52 83 39 13)(7 79 53 84 40 14)(8 80 54 85 33 15)(17 92 59 47 27 70)(18 93 60 48 28 71)(19 94 61 41 29 72)(20 95 62 42 30 65)(21 96 63 43 31 66)(22 89 64 44 32 67)(23 90 57 45 25 68)(24 91 58 46 26 69)
(1 45 86 23)(2 46 87 24)(3 47 88 17)(4 48 81 18)(5 41 82 19)(6 42 83 20)(7 43 84 21)(8 44 85 22)(9 26 56 91)(10 27 49 92)(11 28 50 93)(12 29 51 94)(13 30 52 95)(14 31 53 96)(15 32 54 89)(16 25 55 90)(33 67 80 64)(34 68 73 57)(35 69 74 58)(36 70 75 59)(37 71 76 60)(38 72 77 61)(39 65 78 62)(40 66 79 63)
(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 82)(2 85)(3 88)(4 83)(5 86)(6 81)(7 84)(8 87)(9 33)(10 36)(11 39)(12 34)(13 37)(14 40)(15 35)(16 38)(17 47)(18 42)(19 45)(20 48)(21 43)(22 46)(23 41)(24 44)(25 72)(26 67)(27 70)(28 65)(29 68)(30 71)(31 66)(32 69)(49 75)(50 78)(51 73)(52 76)(53 79)(54 74)(55 77)(56 80)(57 94)(58 89)(59 92)(60 95)(61 90)(62 93)(63 96)(64 91)
G:=sub<Sym(96)| (1,73,55,86,34,16)(2,74,56,87,35,9)(3,75,49,88,36,10)(4,76,50,81,37,11)(5,77,51,82,38,12)(6,78,52,83,39,13)(7,79,53,84,40,14)(8,80,54,85,33,15)(17,92,59,47,27,70)(18,93,60,48,28,71)(19,94,61,41,29,72)(20,95,62,42,30,65)(21,96,63,43,31,66)(22,89,64,44,32,67)(23,90,57,45,25,68)(24,91,58,46,26,69), (1,45,86,23)(2,46,87,24)(3,47,88,17)(4,48,81,18)(5,41,82,19)(6,42,83,20)(7,43,84,21)(8,44,85,22)(9,26,56,91)(10,27,49,92)(11,28,50,93)(12,29,51,94)(13,30,52,95)(14,31,53,96)(15,32,54,89)(16,25,55,90)(33,67,80,64)(34,68,73,57)(35,69,74,58)(36,70,75,59)(37,71,76,60)(38,72,77,61)(39,65,78,62)(40,66,79,63), (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), (1,82)(2,85)(3,88)(4,83)(5,86)(6,81)(7,84)(8,87)(9,33)(10,36)(11,39)(12,34)(13,37)(14,40)(15,35)(16,38)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,72)(26,67)(27,70)(28,65)(29,68)(30,71)(31,66)(32,69)(49,75)(50,78)(51,73)(52,76)(53,79)(54,74)(55,77)(56,80)(57,94)(58,89)(59,92)(60,95)(61,90)(62,93)(63,96)(64,91)>;
G:=Group( (1,73,55,86,34,16)(2,74,56,87,35,9)(3,75,49,88,36,10)(4,76,50,81,37,11)(5,77,51,82,38,12)(6,78,52,83,39,13)(7,79,53,84,40,14)(8,80,54,85,33,15)(17,92,59,47,27,70)(18,93,60,48,28,71)(19,94,61,41,29,72)(20,95,62,42,30,65)(21,96,63,43,31,66)(22,89,64,44,32,67)(23,90,57,45,25,68)(24,91,58,46,26,69), (1,45,86,23)(2,46,87,24)(3,47,88,17)(4,48,81,18)(5,41,82,19)(6,42,83,20)(7,43,84,21)(8,44,85,22)(9,26,56,91)(10,27,49,92)(11,28,50,93)(12,29,51,94)(13,30,52,95)(14,31,53,96)(15,32,54,89)(16,25,55,90)(33,67,80,64)(34,68,73,57)(35,69,74,58)(36,70,75,59)(37,71,76,60)(38,72,77,61)(39,65,78,62)(40,66,79,63), (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), (1,82)(2,85)(3,88)(4,83)(5,86)(6,81)(7,84)(8,87)(9,33)(10,36)(11,39)(12,34)(13,37)(14,40)(15,35)(16,38)(17,47)(18,42)(19,45)(20,48)(21,43)(22,46)(23,41)(24,44)(25,72)(26,67)(27,70)(28,65)(29,68)(30,71)(31,66)(32,69)(49,75)(50,78)(51,73)(52,76)(53,79)(54,74)(55,77)(56,80)(57,94)(58,89)(59,92)(60,95)(61,90)(62,93)(63,96)(64,91) );
G=PermutationGroup([[(1,73,55,86,34,16),(2,74,56,87,35,9),(3,75,49,88,36,10),(4,76,50,81,37,11),(5,77,51,82,38,12),(6,78,52,83,39,13),(7,79,53,84,40,14),(8,80,54,85,33,15),(17,92,59,47,27,70),(18,93,60,48,28,71),(19,94,61,41,29,72),(20,95,62,42,30,65),(21,96,63,43,31,66),(22,89,64,44,32,67),(23,90,57,45,25,68),(24,91,58,46,26,69)], [(1,45,86,23),(2,46,87,24),(3,47,88,17),(4,48,81,18),(5,41,82,19),(6,42,83,20),(7,43,84,21),(8,44,85,22),(9,26,56,91),(10,27,49,92),(11,28,50,93),(12,29,51,94),(13,30,52,95),(14,31,53,96),(15,32,54,89),(16,25,55,90),(33,67,80,64),(34,68,73,57),(35,69,74,58),(36,70,75,59),(37,71,76,60),(38,72,77,61),(39,65,78,62),(40,66,79,63)], [(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)], [(1,82),(2,85),(3,88),(4,83),(5,86),(6,81),(7,84),(8,87),(9,33),(10,36),(11,39),(12,34),(13,37),(14,40),(15,35),(16,38),(17,47),(18,42),(19,45),(20,48),(21,43),(22,46),(23,41),(24,44),(25,72),(26,67),(27,70),(28,65),(29,68),(30,71),(31,66),(32,69),(49,75),(50,78),(51,73),(52,76),(53,79),(54,74),(55,77),(56,80),(57,94),(58,89),(59,92),(60,95),(61,90),(62,93),(63,96),(64,91)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | SD16 | C4oD4 | C4xS3 | C4oD8 | Q8:3S3 | S3xD4 | S3xSD16 | Q8.7D6 |
| kernel | Dic3:8SD16 | C6.D8 | C6.SD16 | C8xDic3 | C3xC4.Q8 | Dic6:C4 | Dic3:5D4 | C2xC24:C2 | C24:C2 | C4.Q8 | C2xDic3 | C4:C4 | C2xC8 | Dic3 | C12 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3:8SD16 ►in GL6(F73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 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 | 72 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 40 | 35 | 0 | 0 | 0 | 0 |
| 21 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 4 | 0 | 0 |
| 0 | 0 | 55 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 44 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 70 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[40,21,0,0,0,0,35,33,0,0,0,0,0,0,61,55,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,44,0,0,0,0,0,1,0,0,0,0,0,0,72,70,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
Dic3:8SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_8{\rm SD}_{16} % in TeX
G:=Group("Dic3:8SD16"); // GroupNames label
G:=SmallGroup(192,411);
// by ID
G=gap.SmallGroup(192,411);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,120,135,268,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations