metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3:3D8, (C2xD8):2S3, (C3xD4):4D4, (C6xD8):12C2, C3:5(C4:D8), (C2xC8).33D6, C2.26(S3xD8), C6.43(C2xD8), C12:3D4:3C2, D4:1(C3:D4), (D4xDic3):4C2, Dic3:C8:17C2, (C2xD4).140D6, C2.D24:16C2, C12.162(C2xD4), C12.90(C4oD4), D4:Dic3:25C2, C4.7(D4:2S3), C2.26(D8:S3), C6.46(C8:C22), (C6xD4).77C22, C22.251(S3xD4), C6.107(C4:D4), (C2xC24).175C22, (C2xC12).427C23, (C2xDic3).179D4, (C2xD12).113C22, C4:Dic3.162C22, (C4xDic3).42C22, C2.22(C23.14D6), (C2xD4:S3):16C2, C4.34(C2xC3:D4), (C2xC6).340(C2xD4), (C2xC3:C8).145C22, (C2xC4).517(C22xS3), SmallGroup(192,709)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3:D8
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=c-1 >
Subgroups: 488 in 140 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, D4, C23, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C2xC8, D8, C22xC4, C2xD4, C2xD4, C3:C8, C24, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C3xD4, C22xS3, C22xC6, D4:C4, C4:C8, C4xD4, C4:1D4, C2xD8, C2xD8, C2xC3:C8, C4xDic3, C4:Dic3, D4:S3, C6.D4, C2xC24, C3xD8, C2xD12, C22xDic3, C2xC3:D4, C6xD4, C4:D8, Dic3:C8, C2.D24, D4:Dic3, C2xD4:S3, D4xDic3, C12:3D4, C6xD8, Dic3:D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, C2xD8, C8:C22, S3xD4, D4:2S3, C2xC3:D4, C4:D8, S3xD8, D8:S3, C23.14D6, Dic3:D8
►On 96 points
(1 88 77 27 51 12)(2 81 78 28 52 13)(3 82 79 29 53 14)(4 83 80 30 54 15)(5 84 73 31 55 16)(6 85 74 32 56 9)(7 86 75 25 49 10)(8 87 76 26 50 11)(17 89 39 63 45 67)(18 90 40 64 46 68)(19 91 33 57 47 69)(20 92 34 58 48 70)(21 93 35 59 41 71)(22 94 36 60 42 72)(23 95 37 61 43 65)(24 96 38 62 44 66)
(1 23 27 61)(2 62 28 24)(3 17 29 63)(4 64 30 18)(5 19 31 57)(6 58 32 20)(7 21 25 59)(8 60 26 22)(9 48 74 92)(10 93 75 41)(11 42 76 94)(12 95 77 43)(13 44 78 96)(14 89 79 45)(15 46 80 90)(16 91 73 47)(33 84 69 55)(34 56 70 85)(35 86 71 49)(36 50 72 87)(37 88 65 51)(38 52 66 81)(39 82 67 53)(40 54 68 83)
(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 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 24)(8 23)(9 67)(10 66)(11 65)(12 72)(13 71)(14 70)(15 69)(16 68)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 64)(32 63)(33 80)(34 79)(35 78)(36 77)(37 76)(38 75)(39 74)(40 73)(41 52)(42 51)(43 50)(44 49)(45 56)(46 55)(47 54)(48 53)(81 93)(82 92)(83 91)(84 90)(85 89)(86 96)(87 95)(88 94)
G:=sub<Sym(96)| (1,88,77,27,51,12)(2,81,78,28,52,13)(3,82,79,29,53,14)(4,83,80,30,54,15)(5,84,73,31,55,16)(6,85,74,32,56,9)(7,86,75,25,49,10)(8,87,76,26,50,11)(17,89,39,63,45,67)(18,90,40,64,46,68)(19,91,33,57,47,69)(20,92,34,58,48,70)(21,93,35,59,41,71)(22,94,36,60,42,72)(23,95,37,61,43,65)(24,96,38,62,44,66), (1,23,27,61)(2,62,28,24)(3,17,29,63)(4,64,30,18)(5,19,31,57)(6,58,32,20)(7,21,25,59)(8,60,26,22)(9,48,74,92)(10,93,75,41)(11,42,76,94)(12,95,77,43)(13,44,78,96)(14,89,79,45)(15,46,80,90)(16,91,73,47)(33,84,69,55)(34,56,70,85)(35,86,71,49)(36,50,72,87)(37,88,65,51)(38,52,66,81)(39,82,67,53)(40,54,68,83), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,24)(8,23)(9,67)(10,66)(11,65)(12,72)(13,71)(14,70)(15,69)(16,68)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,64)(32,63)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,52)(42,51)(43,50)(44,49)(45,56)(46,55)(47,54)(48,53)(81,93)(82,92)(83,91)(84,90)(85,89)(86,96)(87,95)(88,94)>;
G:=Group( (1,88,77,27,51,12)(2,81,78,28,52,13)(3,82,79,29,53,14)(4,83,80,30,54,15)(5,84,73,31,55,16)(6,85,74,32,56,9)(7,86,75,25,49,10)(8,87,76,26,50,11)(17,89,39,63,45,67)(18,90,40,64,46,68)(19,91,33,57,47,69)(20,92,34,58,48,70)(21,93,35,59,41,71)(22,94,36,60,42,72)(23,95,37,61,43,65)(24,96,38,62,44,66), (1,23,27,61)(2,62,28,24)(3,17,29,63)(4,64,30,18)(5,19,31,57)(6,58,32,20)(7,21,25,59)(8,60,26,22)(9,48,74,92)(10,93,75,41)(11,42,76,94)(12,95,77,43)(13,44,78,96)(14,89,79,45)(15,46,80,90)(16,91,73,47)(33,84,69,55)(34,56,70,85)(35,86,71,49)(36,50,72,87)(37,88,65,51)(38,52,66,81)(39,82,67,53)(40,54,68,83), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,24)(8,23)(9,67)(10,66)(11,65)(12,72)(13,71)(14,70)(15,69)(16,68)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,64)(32,63)(33,80)(34,79)(35,78)(36,77)(37,76)(38,75)(39,74)(40,73)(41,52)(42,51)(43,50)(44,49)(45,56)(46,55)(47,54)(48,53)(81,93)(82,92)(83,91)(84,90)(85,89)(86,96)(87,95)(88,94) );
G=PermutationGroup([[(1,88,77,27,51,12),(2,81,78,28,52,13),(3,82,79,29,53,14),(4,83,80,30,54,15),(5,84,73,31,55,16),(6,85,74,32,56,9),(7,86,75,25,49,10),(8,87,76,26,50,11),(17,89,39,63,45,67),(18,90,40,64,46,68),(19,91,33,57,47,69),(20,92,34,58,48,70),(21,93,35,59,41,71),(22,94,36,60,42,72),(23,95,37,61,43,65),(24,96,38,62,44,66)], [(1,23,27,61),(2,62,28,24),(3,17,29,63),(4,64,30,18),(5,19,31,57),(6,58,32,20),(7,21,25,59),(8,60,26,22),(9,48,74,92),(10,93,75,41),(11,42,76,94),(12,95,77,43),(13,44,78,96),(14,89,79,45),(15,46,80,90),(16,91,73,47),(33,84,69,55),(34,56,70,85),(35,86,71,49),(36,50,72,87),(37,88,65,51),(38,52,66,81),(39,82,67,53),(40,54,68,83)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,24),(8,23),(9,67),(10,66),(11,65),(12,72),(13,71),(14,70),(15,69),(16,68),(25,62),(26,61),(27,60),(28,59),(29,58),(30,57),(31,64),(32,63),(33,80),(34,79),(35,78),(36,77),(37,76),(38,75),(39,74),(40,73),(41,52),(42,51),(43,50),(44,49),(45,56),(46,55),(47,54),(48,53),(81,93),(82,92),(83,91),(84,90),(85,89),(86,96),(87,95),(88,94)]])
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 | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 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 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 24 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D8 | C4oD4 | C3:D4 | C8:C22 | D4:2S3 | S3xD4 | S3xD8 | D8:S3 |
| kernel | Dic3:D8 | Dic3:C8 | C2.D24 | D4:Dic3 | C2xD4:S3 | D4xDic3 | C12:3D4 | C6xD8 | C2xD8 | C2xDic3 | C3xD4 | C2xC8 | C2xD4 | Dic3 | C12 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3:D8 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 13 | 30 |
| 0 | 0 | 43 | 60 |
| 0 | 32 | 0 | 0 |
| 57 | 41 | 0 | 0 |
| 0 | 0 | 43 | 60 |
| 0 | 0 | 13 | 30 |
| 0 | 41 | 0 | 0 |
| 57 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,72,0],[72,0,0,0,0,72,0,0,0,0,13,43,0,0,30,60],[0,57,0,0,32,41,0,0,0,0,43,13,0,0,60,30],[0,57,0,0,41,0,0,0,0,0,72,0,0,0,0,72] >;
Dic3:D8 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes D_8 % in TeX
G:=Group("Dic3:D8"); // GroupNames label
G:=SmallGroup(192,709);
// by ID
G=gap.SmallGroup(192,709);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations