metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊1D8, C3⋊C8⋊19D4, C4⋊C4.9D6, C3⋊1(C8⋊7D4), C2.10(S3×D8), C6.24(C2×D8), D4⋊C4⋊2S3, D6⋊3D4⋊1C2, C12⋊D4⋊2C2, (C2×D4).24D6, C4.158(S3×D4), (C2×C8).202D6, C6.Q16⋊6C2, C12.7(C4○D4), C6.40(C4○D8), C2.D24⋊24C2, C12.108(C2×D4), C4.24(C4○D12), C6.15(C4⋊D4), (C2×Dic3).89D4, (C22×S3).47D4, (C6×D4).36C22, C22.173(S3×D4), C2.18(Dic3⋊D4), (C2×C24).227C22, (C2×C12).215C23, (C2×D12).49C22, C4⋊Dic3.70C22, C2.10(Q8.7D6), (S3×C2×C8)⋊18C2, (C2×D4⋊S3)⋊3C2, (C2×C6).228(C2×D4), (C3×D4⋊C4)⋊27C2, (C3×C4⋊C4).17C22, (C2×C3⋊C8).212C22, (S3×C2×C4).224C22, (C2×C4).322(C22×S3), SmallGroup(192,334)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊D8
G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=a4b, dbd=ab, dcd=c-1 >
Subgroups: 472 in 134 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, D4⋊C4, D4⋊C4, C2.D8, C4⋊D4, C22×C8, C2×D8, S3×C8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4⋊S3, C6.D4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C2×D12, C2×C3⋊D4, C6×D4, C8⋊7D4, C6.Q16, C2.D24, C3×D4⋊C4, C12⋊D4, S3×C2×C8, C2×D4⋊S3, D6⋊3D4, D6⋊D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, C22×S3, C4⋊D4, C2×D8, C4○D8, C4○D12, S3×D4, C8⋊7D4, Dic3⋊D4, S3×D8, Q8.7D6, D6⋊D8
►On 96 points
(1 71 49 48 17 74)(2 75 18 41 50 72)(3 65 51 42 19 76)(4 77 20 43 52 66)(5 67 53 44 21 78)(6 79 22 45 54 68)(7 69 55 46 23 80)(8 73 24 47 56 70)(9 28 62 96 88 38)(10 39 81 89 63 29)(11 30 64 90 82 40)(12 33 83 91 57 31)(13 32 58 92 84 34)(14 35 85 93 59 25)(15 26 60 94 86 36)(16 37 87 95 61 27)
(1 29)(2 40)(3 31)(4 34)(5 25)(6 36)(7 27)(8 38)(9 70)(10 74)(11 72)(12 76)(13 66)(14 78)(15 68)(16 80)(17 39)(18 90)(19 33)(20 92)(21 35)(22 94)(23 37)(24 96)(26 54)(28 56)(30 50)(32 52)(41 64)(42 83)(43 58)(44 85)(45 60)(46 87)(47 62)(48 81)(49 89)(51 91)(53 93)(55 95)(57 65)(59 67)(61 69)(63 71)(73 88)(75 82)(77 84)(79 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 90)(10 89)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 49)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 59)(26 58)(27 57)(28 64)(29 63)(30 62)(31 61)(32 60)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 88)(41 47)(42 46)(43 45)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)
G:=sub<Sym(96)| (1,71,49,48,17,74)(2,75,18,41,50,72)(3,65,51,42,19,76)(4,77,20,43,52,66)(5,67,53,44,21,78)(6,79,22,45,54,68)(7,69,55,46,23,80)(8,73,24,47,56,70)(9,28,62,96,88,38)(10,39,81,89,63,29)(11,30,64,90,82,40)(12,33,83,91,57,31)(13,32,58,92,84,34)(14,35,85,93,59,25)(15,26,60,94,86,36)(16,37,87,95,61,27), (1,29)(2,40)(3,31)(4,34)(5,25)(6,36)(7,27)(8,38)(9,70)(10,74)(11,72)(12,76)(13,66)(14,78)(15,68)(16,80)(17,39)(18,90)(19,33)(20,92)(21,35)(22,94)(23,37)(24,96)(26,54)(28,56)(30,50)(32,52)(41,64)(42,83)(43,58)(44,85)(45,60)(46,87)(47,62)(48,81)(49,89)(51,91)(53,93)(55,95)(57,65)(59,67)(61,69)(63,71)(73,88)(75,82)(77,84)(79,86), (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)(9,90)(10,89)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,59)(26,58)(27,57)(28,64)(29,63)(30,62)(31,61)(32,60)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,88)(41,47)(42,46)(43,45)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)>;
G:=Group( (1,71,49,48,17,74)(2,75,18,41,50,72)(3,65,51,42,19,76)(4,77,20,43,52,66)(5,67,53,44,21,78)(6,79,22,45,54,68)(7,69,55,46,23,80)(8,73,24,47,56,70)(9,28,62,96,88,38)(10,39,81,89,63,29)(11,30,64,90,82,40)(12,33,83,91,57,31)(13,32,58,92,84,34)(14,35,85,93,59,25)(15,26,60,94,86,36)(16,37,87,95,61,27), (1,29)(2,40)(3,31)(4,34)(5,25)(6,36)(7,27)(8,38)(9,70)(10,74)(11,72)(12,76)(13,66)(14,78)(15,68)(16,80)(17,39)(18,90)(19,33)(20,92)(21,35)(22,94)(23,37)(24,96)(26,54)(28,56)(30,50)(32,52)(41,64)(42,83)(43,58)(44,85)(45,60)(46,87)(47,62)(48,81)(49,89)(51,91)(53,93)(55,95)(57,65)(59,67)(61,69)(63,71)(73,88)(75,82)(77,84)(79,86), (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)(9,90)(10,89)(11,96)(12,95)(13,94)(14,93)(15,92)(16,91)(17,49)(18,56)(19,55)(20,54)(21,53)(22,52)(23,51)(24,50)(25,59)(26,58)(27,57)(28,64)(29,63)(30,62)(31,61)(32,60)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,88)(41,47)(42,46)(43,45)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73) );
G=PermutationGroup([[(1,71,49,48,17,74),(2,75,18,41,50,72),(3,65,51,42,19,76),(4,77,20,43,52,66),(5,67,53,44,21,78),(6,79,22,45,54,68),(7,69,55,46,23,80),(8,73,24,47,56,70),(9,28,62,96,88,38),(10,39,81,89,63,29),(11,30,64,90,82,40),(12,33,83,91,57,31),(13,32,58,92,84,34),(14,35,85,93,59,25),(15,26,60,94,86,36),(16,37,87,95,61,27)], [(1,29),(2,40),(3,31),(4,34),(5,25),(6,36),(7,27),(8,38),(9,70),(10,74),(11,72),(12,76),(13,66),(14,78),(15,68),(16,80),(17,39),(18,90),(19,33),(20,92),(21,35),(22,94),(23,37),(24,96),(26,54),(28,56),(30,50),(32,52),(41,64),(42,83),(43,58),(44,85),(45,60),(46,87),(47,62),(48,81),(49,89),(51,91),(53,93),(55,95),(57,65),(59,67),(61,69),(63,71),(73,88),(75,82),(77,84),(79,86)], [(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),(9,90),(10,89),(11,96),(12,95),(13,94),(14,93),(15,92),(16,91),(17,49),(18,56),(19,55),(20,54),(21,53),(22,52),(23,51),(24,50),(25,59),(26,58),(27,57),(28,64),(29,63),(30,62),(31,61),(32,60),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,88),(41,47),(42,46),(43,45),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 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 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 24 | 2 | 2 | 2 | 6 | 6 | 8 | 24 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | D8 | C4○D8 | C4○D12 | S3×D4 | S3×D4 | S3×D8 | Q8.7D6 |
| kernel | D6⋊D8 | C6.Q16 | C2.D24 | C3×D4⋊C4 | C12⋊D4 | S3×C2×C8 | C2×D4⋊S3 | D6⋊3D4 | D4⋊C4 | C3⋊C8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | C12 | D6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of D6⋊D8 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 0 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 14 | 66 |
| 16 | 16 | 0 | 0 |
| 57 | 16 | 0 | 0 |
| 0 | 0 | 60 | 43 |
| 0 | 0 | 30 | 13 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,1],[0,27,0,0,46,0,0,0,0,0,7,14,0,0,7,66],[16,57,0,0,16,16,0,0,0,0,60,30,0,0,43,13],[72,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
D6⋊D8 in GAP, Magma, Sage, TeX
D_6\rtimes D_8
% in TeX
G:=Group("D6:D8"); // GroupNames label
G:=SmallGroup(192,334);
// by ID
G=gap.SmallGroup(192,334);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,590,555,297,136,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^4*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations