metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.2D8, D8.5D6, SD32:3S3, C16.11D6, Q16.2D6, C24.19C23, C48.11C22, Dic3.13D8, D24.3C22, Dic12.4C22, C3:C8.14D4, C4.7(S3xD4), (S3xC16):5C2, C3:3(C4oD16), C3:D16:4C2, C48:C2:6C2, C3:Q32:2C2, C2.22(S3xD8), C6.38(C2xD8), D8:3S3:5C2, (C3xSD32):4C2, (C4xS3).21D4, C12.13(C2xD4), C3:C16.6C22, D24:C2:4C2, C8.25(C22xS3), (C3xD8).5C22, (S3xC8).12C22, (C3xQ16).3C22, SmallGroup(192,475)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.2D8
G = < a,b,c,d | a6=b2=d2=1, c8=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c7 >
Subgroups: 300 in 84 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, Q8, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C16, C16, C2xC8, D8, D8, SD16, Q16, Q16, C4oD4, C3:C8, C24, Dic6, C4xS3, C4xS3, D12, C2xDic3, C3:D4, C3xD4, C3xQ8, C2xC16, D16, SD32, SD32, Q32, C4oD8, C3:C16, C48, S3xC8, D24, Dic12, D4.S3, Q8:2S3, C3xD8, C3xQ16, D4:2S3, Q8:3S3, C4oD16, S3xC16, C48:C2, C3:D16, C3:Q32, C3xSD32, D8:3S3, D24:C2, D6.2D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S3xD4, C4oD16, S3xD8, D6.2D8
►On 96 points
Generators in S96
(1 58 44 9 50 36)(2 59 45 10 51 37)(3 60 46 11 52 38)(4 61 47 12 53 39)(5 62 48 13 54 40)(6 63 33 14 55 41)(7 64 34 15 56 42)(8 49 35 16 57 43)(17 71 96 25 79 88)(18 72 81 26 80 89)(19 73 82 27 65 90)(20 74 83 28 66 91)(21 75 84 29 67 92)(22 76 85 30 68 93)(23 77 86 31 69 94)(24 78 87 32 70 95)
(1 30)(2 31)(3 32)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 92)(50 93)(51 94)(52 95)(53 96)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)
(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 75)(2 66)(3 73)(4 80)(5 71)(6 78)(7 69)(8 76)(9 67)(10 74)(11 65)(12 72)(13 79)(14 70)(15 77)(16 68)(17 40)(18 47)(19 38)(20 45)(21 36)(22 43)(23 34)(24 41)(25 48)(26 39)(27 46)(28 37)(29 44)(30 35)(31 42)(32 33)(49 85)(50 92)(51 83)(52 90)(53 81)(54 88)(55 95)(56 86)(57 93)(58 84)(59 91)(60 82)(61 89)(62 96)(63 87)(64 94)
G:=sub<Sym(96)| (1,58,44,9,50,36)(2,59,45,10,51,37)(3,60,46,11,52,38)(4,61,47,12,53,39)(5,62,48,13,54,40)(6,63,33,14,55,41)(7,64,34,15,56,42)(8,49,35,16,57,43)(17,71,96,25,79,88)(18,72,81,26,80,89)(19,73,82,27,65,90)(20,74,83,28,66,91)(21,75,84,29,67,92)(22,76,85,30,68,93)(23,77,86,31,69,94)(24,78,87,32,70,95), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,92)(50,93)(51,94)(52,95)(53,96)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91), (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), (1,75)(2,66)(3,73)(4,80)(5,71)(6,78)(7,69)(8,76)(9,67)(10,74)(11,65)(12,72)(13,79)(14,70)(15,77)(16,68)(17,40)(18,47)(19,38)(20,45)(21,36)(22,43)(23,34)(24,41)(25,48)(26,39)(27,46)(28,37)(29,44)(30,35)(31,42)(32,33)(49,85)(50,92)(51,83)(52,90)(53,81)(54,88)(55,95)(56,86)(57,93)(58,84)(59,91)(60,82)(61,89)(62,96)(63,87)(64,94)>;
G:=Group( (1,58,44,9,50,36)(2,59,45,10,51,37)(3,60,46,11,52,38)(4,61,47,12,53,39)(5,62,48,13,54,40)(6,63,33,14,55,41)(7,64,34,15,56,42)(8,49,35,16,57,43)(17,71,96,25,79,88)(18,72,81,26,80,89)(19,73,82,27,65,90)(20,74,83,28,66,91)(21,75,84,29,67,92)(22,76,85,30,68,93)(23,77,86,31,69,94)(24,78,87,32,70,95), (1,30)(2,31)(3,32)(4,17)(5,18)(6,19)(7,20)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,92)(50,93)(51,94)(52,95)(53,96)(54,81)(55,82)(56,83)(57,84)(58,85)(59,86)(60,87)(61,88)(62,89)(63,90)(64,91), (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), (1,75)(2,66)(3,73)(4,80)(5,71)(6,78)(7,69)(8,76)(9,67)(10,74)(11,65)(12,72)(13,79)(14,70)(15,77)(16,68)(17,40)(18,47)(19,38)(20,45)(21,36)(22,43)(23,34)(24,41)(25,48)(26,39)(27,46)(28,37)(29,44)(30,35)(31,42)(32,33)(49,85)(50,92)(51,83)(52,90)(53,81)(54,88)(55,95)(56,86)(57,93)(58,84)(59,91)(60,82)(61,89)(62,96)(63,87)(64,94) );
G=PermutationGroup([[(1,58,44,9,50,36),(2,59,45,10,51,37),(3,60,46,11,52,38),(4,61,47,12,53,39),(5,62,48,13,54,40),(6,63,33,14,55,41),(7,64,34,15,56,42),(8,49,35,16,57,43),(17,71,96,25,79,88),(18,72,81,26,80,89),(19,73,82,27,65,90),(20,74,83,28,66,91),(21,75,84,29,67,92),(22,76,85,30,68,93),(23,77,86,31,69,94),(24,78,87,32,70,95)], [(1,30),(2,31),(3,32),(4,17),(5,18),(6,19),(7,20),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,92),(50,93),(51,94),(52,95),(53,96),(54,81),(55,82),(56,83),(57,84),(58,85),(59,86),(60,87),(61,88),(62,89),(63,90),(64,91)], [(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)], [(1,75),(2,66),(3,73),(4,80),(5,71),(6,78),(7,69),(8,76),(9,67),(10,74),(11,65),(12,72),(13,79),(14,70),(15,77),(16,68),(17,40),(18,47),(19,38),(20,45),(21,36),(22,43),(23,34),(24,41),(25,48),(26,39),(27,46),(28,37),(29,44),(30,35),(31,42),(32,33),(49,85),(50,92),(51,83),(52,90),(53,81),(54,88),(55,95),(56,86),(57,93),(58,84),(59,91),(60,82),(61,89),(62,96),(63,87),(64,94)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 8A | 8B | 8C | 8D | 12A | 12B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | 24B | 48A | 48B | 48C | 48D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | 24 | 48 | 48 | 48 | 48 |
| size | 1 | 1 | 6 | 8 | 24 | 2 | 2 | 3 | 3 | 8 | 24 | 2 | 16 | 2 | 2 | 6 | 6 | 4 | 16 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 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 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D8 | D8 | C4oD16 | S3xD4 | S3xD8 | D6.2D8 |
| kernel | D6.2D8 | S3xC16 | C48:C2 | C3:D16 | C3:Q32 | C3xSD32 | D8:3S3 | D24:C2 | SD32 | C3:C8 | C4xS3 | C16 | D8 | Q16 | Dic3 | D6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 4 |
Matrix representation of D6.2D8 ►in GL4(F7) generated by
| 0 | 0 | 2 | 5 |
| 2 | 4 | 0 | 4 |
| 5 | 2 | 0 | 4 |
| 2 | 2 | 6 | 5 |
| 4 | 5 | 5 | 5 |
| 4 | 0 | 6 | 0 |
| 2 | 5 | 6 | 6 |
| 5 | 5 | 5 | 4 |
| 1 | 5 | 2 | 2 |
| 4 | 3 | 1 | 2 |
| 6 | 6 | 3 | 2 |
| 1 | 6 | 3 | 5 |
| 0 | 5 | 2 | 6 |
| 2 | 0 | 2 | 1 |
| 3 | 3 | 0 | 1 |
| 1 | 6 | 3 | 0 |
G:=sub<GL(4,GF(7))| [0,2,5,2,0,4,2,2,2,0,0,6,5,4,4,5],[4,4,2,5,5,0,5,5,5,6,6,5,5,0,6,4],[1,4,6,1,5,3,6,6,2,1,3,3,2,2,2,5],[0,2,3,1,5,0,3,6,2,2,0,3,6,1,1,0] >;
D6.2D8 in GAP, Magma, Sage, TeX
D_6._2D_8
% in TeX
G:=Group("D6.2D8"); // GroupNames label
G:=SmallGroup(192,475);
// by ID
G=gap.SmallGroup(192,475);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,758,135,184,346,185,192,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=d^2=1,c^8=a^3,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^7>;
// generators/relations