metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6:3D8, C24:6D4, (C2xD8):5S3, (C6xD8):6C2, C3:5(C8:7D4), C2.29(S3xD8), C6.46(C2xD8), D6:3D4:4C2, C8:10(C3:D4), C24:1C4:23C2, (C2xD4).64D6, (C2xC8).238D6, C6.34(C4oD8), C12.166(C2xD4), C12.93(C4oD4), D4:Dic3:29C2, (C2xC24).90C22, (C22xS3).58D4, (C6xD4).83C22, C22.257(S3xD4), C2.18(D8:3S3), C4.28(D4:2S3), C2.16(D6:3D4), C6.109(C4:D4), (C2xC12).434C23, (C2xDic3).112D4, C4:Dic3.165C22, (S3xC2xC8):3C2, C4.79(C2xC3:D4), (C2xC6).347(C2xD4), (C2xC3:C8).271C22, (S3xC2xC4).239C22, (C2xC4).524(C22xS3), SmallGroup(192,716)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:3D8
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >
Subgroups: 424 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, C23, Dic3, C12, D6, D6, C2xC6, C2xC6, C22:C4, C4:C4, C2xC8, C2xC8, D8, C22xC4, C2xD4, C2xD4, C3:C8, C24, C4xS3, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, D4:C4, C2.D8, C4:D4, C22xC8, C2xD8, S3xC8, C2xC3:C8, C4:Dic3, C6.D4, C2xC24, C3xD8, S3xC2xC4, C2xC3:D4, C6xD4, C8:7D4, C24:1C4, D4:Dic3, S3xC2xC8, D6:3D4, C6xD8, D6:3D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C4oD4, C3:D4, C22xS3, C4:D4, C2xD8, C4oD8, S3xD4, D4:2S3, C2xC3:D4, C8:7D4, S3xD8, D8:3S3, D6:3D4, D6:3D8
►On 96 points
(1 47 17 75 50 88)(2 48 18 76 51 81)(3 41 19 77 52 82)(4 42 20 78 53 83)(5 43 21 79 54 84)(6 44 22 80 55 85)(7 45 23 73 56 86)(8 46 24 74 49 87)(9 72 91 58 30 35)(10 65 92 59 31 36)(11 66 93 60 32 37)(12 67 94 61 25 38)(13 68 95 62 26 39)(14 69 96 63 27 40)(15 70 89 64 28 33)(16 71 90 57 29 34)
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 89)(8 90)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(25 53)(26 54)(27 55)(28 56)(29 49)(30 50)(31 51)(32 52)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(57 87)(58 88)(59 81)(60 82)(61 83)(62 84)(63 85)(64 86)
(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)
(2 8)(3 7)(4 6)(9 58)(10 57)(11 64)(12 63)(13 62)(14 61)(15 60)(16 59)(18 24)(19 23)(20 22)(25 69)(26 68)(27 67)(28 66)(29 65)(30 72)(31 71)(32 70)(33 93)(34 92)(35 91)(36 90)(37 89)(38 96)(39 95)(40 94)(41 45)(42 44)(46 48)(49 51)(52 56)(53 55)(73 77)(74 76)(78 80)(81 87)(82 86)(83 85)
G:=sub<Sym(96)| (1,47,17,75,50,88)(2,48,18,76,51,81)(3,41,19,77,52,82)(4,42,20,78,53,83)(5,43,21,79,54,84)(6,44,22,80,55,85)(7,45,23,73,56,86)(8,46,24,74,49,87)(9,72,91,58,30,35)(10,65,92,59,31,36)(11,66,93,60,32,37)(12,67,94,61,25,38)(13,68,95,62,26,39)(14,69,96,63,27,40)(15,70,89,64,28,33)(16,71,90,57,29,34), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,89)(8,90)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(57,87)(58,88)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86), (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), (2,8)(3,7)(4,6)(9,58)(10,57)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(18,24)(19,23)(20,22)(25,69)(26,68)(27,67)(28,66)(29,65)(30,72)(31,71)(32,70)(33,93)(34,92)(35,91)(36,90)(37,89)(38,96)(39,95)(40,94)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(73,77)(74,76)(78,80)(81,87)(82,86)(83,85)>;
G:=Group( (1,47,17,75,50,88)(2,48,18,76,51,81)(3,41,19,77,52,82)(4,42,20,78,53,83)(5,43,21,79,54,84)(6,44,22,80,55,85)(7,45,23,73,56,86)(8,46,24,74,49,87)(9,72,91,58,30,35)(10,65,92,59,31,36)(11,66,93,60,32,37)(12,67,94,61,25,38)(13,68,95,62,26,39)(14,69,96,63,27,40)(15,70,89,64,28,33)(16,71,90,57,29,34), (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,89)(8,90)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(25,53)(26,54)(27,55)(28,56)(29,49)(30,50)(31,51)(32,52)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(57,87)(58,88)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86), (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), (2,8)(3,7)(4,6)(9,58)(10,57)(11,64)(12,63)(13,62)(14,61)(15,60)(16,59)(18,24)(19,23)(20,22)(25,69)(26,68)(27,67)(28,66)(29,65)(30,72)(31,71)(32,70)(33,93)(34,92)(35,91)(36,90)(37,89)(38,96)(39,95)(40,94)(41,45)(42,44)(46,48)(49,51)(52,56)(53,55)(73,77)(74,76)(78,80)(81,87)(82,86)(83,85) );
G=PermutationGroup([[(1,47,17,75,50,88),(2,48,18,76,51,81),(3,41,19,77,52,82),(4,42,20,78,53,83),(5,43,21,79,54,84),(6,44,22,80,55,85),(7,45,23,73,56,86),(8,46,24,74,49,87),(9,72,91,58,30,35),(10,65,92,59,31,36),(11,66,93,60,32,37),(12,67,94,61,25,38),(13,68,95,62,26,39),(14,69,96,63,27,40),(15,70,89,64,28,33),(16,71,90,57,29,34)], [(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,89),(8,90),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(25,53),(26,54),(27,55),(28,56),(29,49),(30,50),(31,51),(32,52),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,65),(57,87),(58,88),(59,81),(60,82),(61,83),(62,84),(63,85),(64,86)], [(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)], [(2,8),(3,7),(4,6),(9,58),(10,57),(11,64),(12,63),(13,62),(14,61),(15,60),(16,59),(18,24),(19,23),(20,22),(25,69),(26,68),(27,67),(28,66),(29,65),(30,72),(31,71),(32,70),(33,93),(34,92),(35,91),(36,90),(37,89),(38,96),(39,95),(40,94),(41,45),(42,44),(46,48),(49,51),(52,56),(53,55),(73,77),(74,76),(78,80),(81,87),(82,86),(83,85)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 8 | 2 | 2 | 2 | 6 | 6 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C4oD4 | D8 | C3:D4 | C4oD8 | D4:2S3 | S3xD4 | S3xD8 | D8:3S3 |
| kernel | D6:3D8 | C24:1C4 | D4:Dic3 | S3xC2xC8 | D6:3D4 | C6xD8 | C2xD8 | C24 | C2xDic3 | C22xS3 | C2xC8 | C2xD4 | C12 | D6 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of D6:3D8 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 12 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 49 | 39 |
| 0 | 0 | 62 | 24 |
| 16 | 16 | 0 | 0 |
| 57 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 63 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,65,12,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,49,62,0,0,39,24],[16,57,0,0,16,16,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,72,0,0,0,0,1,63,0,0,0,72] >;
D6:3D8 in GAP, Magma, Sage, TeX
D_6\rtimes_3D_8
% in TeX
G:=Group("D6:3D8"); // GroupNames label
G:=SmallGroup(192,716);
// by ID
G=gap.SmallGroup(192,716);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations