metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.4D8, D6⋊C8⋊5C2, (C2×C8).9D6, C2.9(S3×D8), C24⋊1C4⋊8C2, C6.23(C2×D8), D4⋊C4⋊5S3, C4⋊C4.135D6, (C2×D4).23D6, C6.Q16⋊5C2, D6⋊3D4.1C2, D4⋊Dic3⋊7C2, (C2×C24).9C22, C4.51(C4○D12), (C2×Dic3).19D4, (C22×S3).71D4, (C6×D4).35C22, C22.172(S3×D4), C3⋊2(C22.D8), C12.149(C4○D4), C4.78(D4⋊2S3), (C2×C12).214C23, C2.11(D4.D6), C6.29(C8.C22), C4⋊Dic3.69C22, C2.12(C23.9D6), C6.20(C22.D4), (S3×C4⋊C4)⋊3C2, (S3×C2×C4).8C22, (C3×D4⋊C4)⋊5C2, (C2×C6).227(C2×D4), (C2×C3⋊C8).15C22, (C3×C4⋊C4).16C22, (C2×C4).321(C22×S3), SmallGroup(192,333)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.D8
G = < a,b,c,d | a6=b2=c8=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c-1 >
Subgroups: 360 in 114 conjugacy classes, 39 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, C22×C4, C2×D4, C2×D4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22⋊C8, D4⋊C4, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×C3⋊D4, C6×D4, C22.D8, C6.Q16, C24⋊1C4, D6⋊C8, D4⋊Dic3, C3×D4⋊C4, S3×C4⋊C4, D6⋊3D4, D6.D8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, C22×S3, C22.D4, C2×D8, C8.C22, C4○D12, S3×D4, D4⋊2S3, C22.D8, C23.9D6, S3×D8, D4.D6, D6.D8
►On 96 points
(1 55 77 85 38 16)(2 56 78 86 39 9)(3 49 79 87 40 10)(4 50 80 88 33 11)(5 51 73 81 34 12)(6 52 74 82 35 13)(7 53 75 83 36 14)(8 54 76 84 37 15)(17 64 93 67 28 48)(18 57 94 68 29 41)(19 58 95 69 30 42)(20 59 96 70 31 43)(21 60 89 71 32 44)(22 61 90 72 25 45)(23 62 91 65 26 46)(24 63 92 66 27 47)
(1 68)(2 19)(3 70)(4 21)(5 72)(6 23)(7 66)(8 17)(9 58)(10 31)(11 60)(12 25)(13 62)(14 27)(15 64)(16 29)(18 85)(20 87)(22 81)(24 83)(26 74)(28 76)(30 78)(32 80)(33 89)(34 45)(35 91)(36 47)(37 93)(38 41)(39 95)(40 43)(42 56)(44 50)(46 52)(48 54)(49 96)(51 90)(53 92)(55 94)(57 77)(59 79)(61 73)(63 75)(65 82)(67 84)(69 86)(71 88)
(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 84 85 8)(2 7 86 83)(3 82 87 6)(4 5 88 81)(9 14 78 75)(10 74 79 13)(11 12 80 73)(15 16 76 77)(17 68 67 18)(19 66 69 24)(20 23 70 65)(21 72 71 22)(25 32 61 60)(26 59 62 31)(27 30 63 58)(28 57 64 29)(33 34 50 51)(35 40 52 49)(36 56 53 39)(37 38 54 55)(41 48 94 93)(42 92 95 47)(43 46 96 91)(44 90 89 45)
G:=sub<Sym(96)| (1,55,77,85,38,16)(2,56,78,86,39,9)(3,49,79,87,40,10)(4,50,80,88,33,11)(5,51,73,81,34,12)(6,52,74,82,35,13)(7,53,75,83,36,14)(8,54,76,84,37,15)(17,64,93,67,28,48)(18,57,94,68,29,41)(19,58,95,69,30,42)(20,59,96,70,31,43)(21,60,89,71,32,44)(22,61,90,72,25,45)(23,62,91,65,26,46)(24,63,92,66,27,47), (1,68)(2,19)(3,70)(4,21)(5,72)(6,23)(7,66)(8,17)(9,58)(10,31)(11,60)(12,25)(13,62)(14,27)(15,64)(16,29)(18,85)(20,87)(22,81)(24,83)(26,74)(28,76)(30,78)(32,80)(33,89)(34,45)(35,91)(36,47)(37,93)(38,41)(39,95)(40,43)(42,56)(44,50)(46,52)(48,54)(49,96)(51,90)(53,92)(55,94)(57,77)(59,79)(61,73)(63,75)(65,82)(67,84)(69,86)(71,88), (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), (1,84,85,8)(2,7,86,83)(3,82,87,6)(4,5,88,81)(9,14,78,75)(10,74,79,13)(11,12,80,73)(15,16,76,77)(17,68,67,18)(19,66,69,24)(20,23,70,65)(21,72,71,22)(25,32,61,60)(26,59,62,31)(27,30,63,58)(28,57,64,29)(33,34,50,51)(35,40,52,49)(36,56,53,39)(37,38,54,55)(41,48,94,93)(42,92,95,47)(43,46,96,91)(44,90,89,45)>;
G:=Group( (1,55,77,85,38,16)(2,56,78,86,39,9)(3,49,79,87,40,10)(4,50,80,88,33,11)(5,51,73,81,34,12)(6,52,74,82,35,13)(7,53,75,83,36,14)(8,54,76,84,37,15)(17,64,93,67,28,48)(18,57,94,68,29,41)(19,58,95,69,30,42)(20,59,96,70,31,43)(21,60,89,71,32,44)(22,61,90,72,25,45)(23,62,91,65,26,46)(24,63,92,66,27,47), (1,68)(2,19)(3,70)(4,21)(5,72)(6,23)(7,66)(8,17)(9,58)(10,31)(11,60)(12,25)(13,62)(14,27)(15,64)(16,29)(18,85)(20,87)(22,81)(24,83)(26,74)(28,76)(30,78)(32,80)(33,89)(34,45)(35,91)(36,47)(37,93)(38,41)(39,95)(40,43)(42,56)(44,50)(46,52)(48,54)(49,96)(51,90)(53,92)(55,94)(57,77)(59,79)(61,73)(63,75)(65,82)(67,84)(69,86)(71,88), (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), (1,84,85,8)(2,7,86,83)(3,82,87,6)(4,5,88,81)(9,14,78,75)(10,74,79,13)(11,12,80,73)(15,16,76,77)(17,68,67,18)(19,66,69,24)(20,23,70,65)(21,72,71,22)(25,32,61,60)(26,59,62,31)(27,30,63,58)(28,57,64,29)(33,34,50,51)(35,40,52,49)(36,56,53,39)(37,38,54,55)(41,48,94,93)(42,92,95,47)(43,46,96,91)(44,90,89,45) );
G=PermutationGroup([[(1,55,77,85,38,16),(2,56,78,86,39,9),(3,49,79,87,40,10),(4,50,80,88,33,11),(5,51,73,81,34,12),(6,52,74,82,35,13),(7,53,75,83,36,14),(8,54,76,84,37,15),(17,64,93,67,28,48),(18,57,94,68,29,41),(19,58,95,69,30,42),(20,59,96,70,31,43),(21,60,89,71,32,44),(22,61,90,72,25,45),(23,62,91,65,26,46),(24,63,92,66,27,47)], [(1,68),(2,19),(3,70),(4,21),(5,72),(6,23),(7,66),(8,17),(9,58),(10,31),(11,60),(12,25),(13,62),(14,27),(15,64),(16,29),(18,85),(20,87),(22,81),(24,83),(26,74),(28,76),(30,78),(32,80),(33,89),(34,45),(35,91),(36,47),(37,93),(38,41),(39,95),(40,43),(42,56),(44,50),(46,52),(48,54),(49,96),(51,90),(53,92),(55,94),(57,77),(59,79),(61,73),(63,75),(65,82),(67,84),(69,86),(71,88)], [(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)], [(1,84,85,8),(2,7,86,83),(3,82,87,6),(4,5,88,81),(9,14,78,75),(10,74,79,13),(11,12,80,73),(15,16,76,77),(17,68,67,18),(19,66,69,24),(20,23,70,65),(21,72,71,22),(25,32,61,60),(26,59,62,31),(27,30,63,58),(28,57,64,29),(33,34,50,51),(35,40,52,49),(36,56,53,39),(37,38,54,55),(41,48,94,93),(42,92,95,47),(43,46,96,91),(44,90,89,45)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | D8 | C4○D12 | C8.C22 | D4⋊2S3 | S3×D4 | S3×D8 | D4.D6 |
| kernel | D6.D8 | C6.Q16 | C24⋊1C4 | D6⋊C8 | D4⋊Dic3 | C3×D4⋊C4 | S3×C4⋊C4 | D6⋊3D4 | D4⋊C4 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | C12 | D6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6.D8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 64 | 9 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 9 | 56 |
| 0 | 0 | 9 | 64 |
| 16 | 57 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 53 | 27 |
| 16 | 57 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,65,64,0,0,0,9],[72,0,0,0,0,72,0,0,0,0,9,9,0,0,56,64],[16,16,0,0,57,16,0,0,0,0,46,53,0,0,0,27],[16,57,0,0,57,57,0,0,0,0,46,0,0,0,0,46] >;
D6.D8 in GAP, Magma, Sage, TeX
D_6.D_8
% in TeX
G:=Group("D6.D8"); // GroupNames label
G:=SmallGroup(192,333);
// by ID
G=gap.SmallGroup(192,333);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,254,219,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=a^3*c^-1>;
// generators/relations