metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊2D8, C24⋊4D4, C8⋊7D12, C2.D8⋊3S3, C3⋊2(C8⋊7D4), C4⋊C4.48D6, C6.30(C2×D8), C2.14(S3×D8), (C2×D24)⋊16C2, C12⋊D4⋊7C2, C4.53(C2×D12), (C2×C8).231D6, C6.75(C4○D8), C12.133(C2×D4), C6.D8⋊21C2, C12.38(C4○D4), C6.46(C4⋊D4), (C2×C24).83C22, C4.9(Q8⋊3S3), (C22×S3).55D4, C22.229(S3×D4), C2.19(C12⋊D4), (C2×C12).299C23, (C2×Dic3).103D4, (C2×D12).82C22, C2.13(D24⋊C2), (S3×C2×C8)⋊2C2, (C3×C2.D8)⋊5C2, (C2×C6).304(C2×D4), (C3×C4⋊C4).92C22, (C2×C3⋊C8).235C22, (S3×C2×C4).236C22, (C2×C4).402(C22×S3), SmallGroup(192,442)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊2D8
G = < a,b,c,d | a6=b2=c8=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Subgroups: 512 in 134 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, S3×C8, D24, C2×C3⋊C8, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C2×D12, C8⋊7D4, C6.D8, C3×C2.D8, C12⋊D4, S3×C2×C8, C2×D24, D6⋊2D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D8, C4○D8, C2×D12, S3×D4, Q8⋊3S3, C8⋊7D4, C12⋊D4, S3×D8, D24⋊C2, D6⋊2D8
►On 96 points
Generators in S96
(1 31 71 45 82 95)(2 32 72 46 83 96)(3 25 65 47 84 89)(4 26 66 48 85 90)(5 27 67 41 86 91)(6 28 68 42 87 92)(7 29 69 43 88 93)(8 30 70 44 81 94)(9 61 40 20 77 52)(10 62 33 21 78 53)(11 63 34 22 79 54)(12 64 35 23 80 55)(13 57 36 24 73 56)(14 58 37 17 74 49)(15 59 38 18 75 50)(16 60 39 19 76 51)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 81)(24 82)(33 92)(34 93)(35 94)(36 95)(37 96)(38 89)(39 90)(40 91)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(49 72)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)
(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 4)(2 3)(5 8)(6 7)(9 64)(10 63)(11 62)(12 61)(13 60)(14 59)(15 58)(16 57)(17 75)(18 74)(19 73)(20 80)(21 79)(22 78)(23 77)(24 76)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 90)(32 89)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 56)(40 55)(41 44)(42 43)(45 48)(46 47)(65 83)(66 82)(67 81)(68 88)(69 87)(70 86)(71 85)(72 84)
G:=sub<Sym(96)| (1,31,71,45,82,95)(2,32,72,46,83,96)(3,25,65,47,84,89)(4,26,66,48,85,90)(5,27,67,41,86,91)(6,28,68,42,87,92)(7,29,69,43,88,93)(8,30,70,44,81,94)(9,61,40,20,77,52)(10,62,33,21,78,53)(11,63,34,22,79,54)(12,64,35,23,80,55)(13,57,36,24,73,56)(14,58,37,17,74,49)(15,59,38,18,75,50)(16,60,39,19,76,51), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(33,92)(34,93)(35,94)(36,95)(37,96)(38,89)(39,90)(40,91)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (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), (1,4)(2,3)(5,8)(6,7)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,75)(18,74)(19,73)(20,80)(21,79)(22,78)(23,77)(24,76)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,56)(40,55)(41,44)(42,43)(45,48)(46,47)(65,83)(66,82)(67,81)(68,88)(69,87)(70,86)(71,85)(72,84)>;
G:=Group( (1,31,71,45,82,95)(2,32,72,46,83,96)(3,25,65,47,84,89)(4,26,66,48,85,90)(5,27,67,41,86,91)(6,28,68,42,87,92)(7,29,69,43,88,93)(8,30,70,44,81,94)(9,61,40,20,77,52)(10,62,33,21,78,53)(11,63,34,22,79,54)(12,64,35,23,80,55)(13,57,36,24,73,56)(14,58,37,17,74,49)(15,59,38,18,75,50)(16,60,39,19,76,51), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,81)(24,82)(33,92)(34,93)(35,94)(36,95)(37,96)(38,89)(39,90)(40,91)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(49,72)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71), (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), (1,4)(2,3)(5,8)(6,7)(9,64)(10,63)(11,62)(12,61)(13,60)(14,59)(15,58)(16,57)(17,75)(18,74)(19,73)(20,80)(21,79)(22,78)(23,77)(24,76)(25,96)(26,95)(27,94)(28,93)(29,92)(30,91)(31,90)(32,89)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,56)(40,55)(41,44)(42,43)(45,48)(46,47)(65,83)(66,82)(67,81)(68,88)(69,87)(70,86)(71,85)(72,84) );
G=PermutationGroup([[(1,31,71,45,82,95),(2,32,72,46,83,96),(3,25,65,47,84,89),(4,26,66,48,85,90),(5,27,67,41,86,91),(6,28,68,42,87,92),(7,29,69,43,88,93),(8,30,70,44,81,94),(9,61,40,20,77,52),(10,62,33,21,78,53),(11,63,34,22,79,54),(12,64,35,23,80,55),(13,57,36,24,73,56),(14,58,37,17,74,49),(15,59,38,18,75,50),(16,60,39,19,76,51)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26),(17,83),(18,84),(19,85),(20,86),(21,87),(22,88),(23,81),(24,82),(33,92),(34,93),(35,94),(36,95),(37,96),(38,89),(39,90),(40,91),(41,77),(42,78),(43,79),(44,80),(45,73),(46,74),(47,75),(48,76),(49,72),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71)], [(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)], [(1,4),(2,3),(5,8),(6,7),(9,64),(10,63),(11,62),(12,61),(13,60),(14,59),(15,58),(16,57),(17,75),(18,74),(19,73),(20,80),(21,79),(22,78),(23,77),(24,76),(25,96),(26,95),(27,94),(28,93),(29,92),(30,91),(31,90),(32,89),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,56),(40,55),(41,44),(42,43),(45,48),(46,47),(65,83),(66,82),(67,81),(68,88),(69,87),(70,86),(71,85),(72,84)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 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 | 2 | 2 | 3 | 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 | 6 | 6 | 24 | 24 | 2 | 2 | 2 | 6 | 6 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | C4○D4 | D8 | D12 | C4○D8 | Q8⋊3S3 | S3×D4 | S3×D8 | D24⋊C2 |
| kernel | D6⋊2D8 | C6.D8 | C3×C2.D8 | C12⋊D4 | S3×C2×C8 | C2×D24 | C2.D8 | C24 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C12 | D6 | C8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of D6⋊2D8 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 7 | 66 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 27 | 8 |
| 0 | 0 | 55 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 41 | 25 |
| 0 | 0 | 35 | 0 |
| 1 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 32 | 48 |
| 0 | 0 | 38 | 41 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[7,59,0,0,66,66,0,0,0,0,27,55,0,0,8,46],[1,0,0,0,0,1,0,0,0,0,41,35,0,0,25,0],[1,72,0,0,0,72,0,0,0,0,32,38,0,0,48,41] >;
D6⋊2D8 in GAP, Magma, Sage, TeX
D_6\rtimes_2D_8
% in TeX
G:=Group("D6:2D8"); // GroupNames label
G:=SmallGroup(192,442);
// by ID
G=gap.SmallGroup(192,442);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations