metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C8.4C2, (C2×C8).19D6, C4⋊C4.153D6, Q8⋊C4⋊7S3, C24⋊1C4⋊11C2, (C2×Q8).44D6, C6.71(C4○D8), Q8⋊2Dic3⋊9C2, D6⋊3Q8.4C2, C4.57(C4○D12), (C2×C24).19C22, C12.Q8⋊12C2, (C22×S3).20D4, C22.204(S3×D4), C12.163(C4○D4), (C6×Q8).37C22, C4.88(D4⋊2S3), (C2×C12).254C23, (C2×Dic3).156D4, C2.18(D4.D6), C6.36(C8.C22), C3⋊2(C23.20D4), C4⋊Dic3.98C22, C2.10(D24⋊C2), C2.18(C23.9D6), C6.26(C22.D4), C4⋊C4⋊7S3.2C2, (C3×Q8⋊C4)⋊7C2, (C2×C6).267(C2×D4), (C2×C3⋊C8).44C22, (S3×C2×C4).26C22, (C3×C4⋊C4).55C22, (C2×C4).361(C22×S3), SmallGroup(192,373)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C8.C2
G = < a,b,c,d | a6=b2=c8=1, d2=a3c4, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd-1=bc4, dcd-1=a3c-1 >
Subgroups: 264 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C6×Q8, C23.20D4, C12.Q8, C24⋊1C4, D6⋊C8, Q8⋊2Dic3, C3×Q8⋊C4, C4⋊C4⋊7S3, D6⋊3Q8, D6⋊C8.C2
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D8, C8.C22, C4○D12, S3×D4, D4⋊2S3, C23.20D4, C23.9D6, D4.D6, D24⋊C2, D6⋊C8.C2
►On 96 points
(1 61 39 23 93 75)(2 62 40 24 94 76)(3 63 33 17 95 77)(4 64 34 18 96 78)(5 57 35 19 89 79)(6 58 36 20 90 80)(7 59 37 21 91 73)(8 60 38 22 92 74)(9 83 67 32 55 47)(10 84 68 25 56 48)(11 85 69 26 49 41)(12 86 70 27 50 42)(13 87 71 28 51 43)(14 88 72 29 52 44)(15 81 65 30 53 45)(16 82 66 31 54 46)
(1 79)(2 36)(3 73)(4 38)(5 75)(6 40)(7 77)(8 34)(9 67)(10 48)(11 69)(12 42)(13 71)(14 44)(15 65)(16 46)(17 37)(18 74)(19 39)(20 76)(21 33)(22 78)(23 35)(24 80)(25 68)(26 41)(27 70)(28 43)(29 72)(30 45)(31 66)(32 47)(50 86)(52 88)(54 82)(56 84)(57 93)(58 62)(59 95)(60 64)(61 89)(63 91)(90 94)(92 96)
(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 12 19 31)(2 26 20 15)(3 10 21 29)(4 32 22 13)(5 16 23 27)(6 30 24 11)(7 14 17 25)(8 28 18 9)(33 68 73 44)(34 47 74 71)(35 66 75 42)(36 45 76 69)(37 72 77 48)(38 43 78 67)(39 70 79 46)(40 41 80 65)(49 90 81 62)(50 57 82 93)(51 96 83 60)(52 63 84 91)(53 94 85 58)(54 61 86 89)(55 92 87 64)(56 59 88 95)
G:=sub<Sym(96)| (1,61,39,23,93,75)(2,62,40,24,94,76)(3,63,33,17,95,77)(4,64,34,18,96,78)(5,57,35,19,89,79)(6,58,36,20,90,80)(7,59,37,21,91,73)(8,60,38,22,92,74)(9,83,67,32,55,47)(10,84,68,25,56,48)(11,85,69,26,49,41)(12,86,70,27,50,42)(13,87,71,28,51,43)(14,88,72,29,52,44)(15,81,65,30,53,45)(16,82,66,31,54,46), (1,79)(2,36)(3,73)(4,38)(5,75)(6,40)(7,77)(8,34)(9,67)(10,48)(11,69)(12,42)(13,71)(14,44)(15,65)(16,46)(17,37)(18,74)(19,39)(20,76)(21,33)(22,78)(23,35)(24,80)(25,68)(26,41)(27,70)(28,43)(29,72)(30,45)(31,66)(32,47)(50,86)(52,88)(54,82)(56,84)(57,93)(58,62)(59,95)(60,64)(61,89)(63,91)(90,94)(92,96), (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), (1,12,19,31)(2,26,20,15)(3,10,21,29)(4,32,22,13)(5,16,23,27)(6,30,24,11)(7,14,17,25)(8,28,18,9)(33,68,73,44)(34,47,74,71)(35,66,75,42)(36,45,76,69)(37,72,77,48)(38,43,78,67)(39,70,79,46)(40,41,80,65)(49,90,81,62)(50,57,82,93)(51,96,83,60)(52,63,84,91)(53,94,85,58)(54,61,86,89)(55,92,87,64)(56,59,88,95)>;
G:=Group( (1,61,39,23,93,75)(2,62,40,24,94,76)(3,63,33,17,95,77)(4,64,34,18,96,78)(5,57,35,19,89,79)(6,58,36,20,90,80)(7,59,37,21,91,73)(8,60,38,22,92,74)(9,83,67,32,55,47)(10,84,68,25,56,48)(11,85,69,26,49,41)(12,86,70,27,50,42)(13,87,71,28,51,43)(14,88,72,29,52,44)(15,81,65,30,53,45)(16,82,66,31,54,46), (1,79)(2,36)(3,73)(4,38)(5,75)(6,40)(7,77)(8,34)(9,67)(10,48)(11,69)(12,42)(13,71)(14,44)(15,65)(16,46)(17,37)(18,74)(19,39)(20,76)(21,33)(22,78)(23,35)(24,80)(25,68)(26,41)(27,70)(28,43)(29,72)(30,45)(31,66)(32,47)(50,86)(52,88)(54,82)(56,84)(57,93)(58,62)(59,95)(60,64)(61,89)(63,91)(90,94)(92,96), (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), (1,12,19,31)(2,26,20,15)(3,10,21,29)(4,32,22,13)(5,16,23,27)(6,30,24,11)(7,14,17,25)(8,28,18,9)(33,68,73,44)(34,47,74,71)(35,66,75,42)(36,45,76,69)(37,72,77,48)(38,43,78,67)(39,70,79,46)(40,41,80,65)(49,90,81,62)(50,57,82,93)(51,96,83,60)(52,63,84,91)(53,94,85,58)(54,61,86,89)(55,92,87,64)(56,59,88,95) );
G=PermutationGroup([[(1,61,39,23,93,75),(2,62,40,24,94,76),(3,63,33,17,95,77),(4,64,34,18,96,78),(5,57,35,19,89,79),(6,58,36,20,90,80),(7,59,37,21,91,73),(8,60,38,22,92,74),(9,83,67,32,55,47),(10,84,68,25,56,48),(11,85,69,26,49,41),(12,86,70,27,50,42),(13,87,71,28,51,43),(14,88,72,29,52,44),(15,81,65,30,53,45),(16,82,66,31,54,46)], [(1,79),(2,36),(3,73),(4,38),(5,75),(6,40),(7,77),(8,34),(9,67),(10,48),(11,69),(12,42),(13,71),(14,44),(15,65),(16,46),(17,37),(18,74),(19,39),(20,76),(21,33),(22,78),(23,35),(24,80),(25,68),(26,41),(27,70),(28,43),(29,72),(30,45),(31,66),(32,47),(50,86),(52,88),(54,82),(56,84),(57,93),(58,62),(59,95),(60,64),(61,89),(63,91),(90,94),(92,96)], [(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)], [(1,12,19,31),(2,26,20,15),(3,10,21,29),(4,32,22,13),(5,16,23,27),(6,30,24,11),(7,14,17,25),(8,28,18,9),(33,68,73,44),(34,47,74,71),(35,66,75,42),(36,45,76,69),(37,72,77,48),(38,43,78,67),(39,70,79,46),(40,41,80,65),(49,90,81,62),(50,57,82,93),(51,96,83,60),(52,63,84,91),(53,94,85,58),(54,61,86,89),(55,92,87,64),(56,59,88,95)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 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 | C4○D8 | C4○D12 | C8.C22 | D4⋊2S3 | S3×D4 | D4.D6 | D24⋊C2 |
| kernel | D6⋊C8.C2 | C12.Q8 | C24⋊1C4 | D6⋊C8 | Q8⋊2Dic3 | C3×Q8⋊C4 | C4⋊C4⋊7S3 | D6⋊3Q8 | Q8⋊C4 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | C12 | C6 | 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⋊C8.C2 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 1 |
| 72 | 0 | 0 | 0 |
| 22 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 10 | 0 | 0 | 0 |
| 59 | 22 | 0 | 0 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 59 | 66 |
| 4 | 7 | 0 | 0 |
| 8 | 69 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,1],[72,22,0,0,0,1,0,0,0,0,0,72,0,0,72,0],[10,59,0,0,0,22,0,0,0,0,7,59,0,0,14,66],[4,8,0,0,7,69,0,0,0,0,46,0,0,0,0,46] >;
D6⋊C8.C2 in GAP, Magma, Sage, TeX
D_6\rtimes C_8.C_2
% in TeX
G:=Group("D6:C8.C2"); // GroupNames label
G:=SmallGroup(192,373);
// by ID
G=gap.SmallGroup(192,373);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,254,219,184,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3*c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,d*b*d^-1=b*c^4,d*c*d^-1=a^3*c^-1>;
// generators/relations