metabelian, supersoluble, monomial
Aliases: D6⋊2D12, C62.78C23, (S3×C6)⋊9D4, (C3×C12)⋊4D4, (C6×D12)⋊5C2, (C2×D12)⋊4S3, C6.21(S3×D4), C12⋊9(C3⋊D4), D6⋊Dic3⋊4C2, C6.22(C2×D12), C2.23(S3×D12), C3⋊3(D6⋊3D4), C4⋊2(D6⋊S3), C3⋊3(C12⋊7D4), (C2×C12).282D6, C32⋊6(C4⋊D4), C6.48(C4○D12), (C22×S3).14D6, C12⋊Dic3⋊19C2, C6.27(D4⋊2S3), (C6×C12).105C22, (C2×Dic3).100D6, C2.18(D12⋊5S3), (C6×Dic3).115C22, (S3×C2×C4)⋊1S3, (S3×C2×C12)⋊3C2, (C2×C4).81S32, (C3×C6).57(C2×D4), C6.79(C2×C3⋊D4), (C2×D6⋊S3)⋊3C2, C22.116(C2×S32), (S3×C2×C6).29C22, (C3×C6).48(C4○D4), C2.13(C2×D6⋊S3), (C2×C6).97(C22×S3), (C2×C3⋊Dic3).55C22, SmallGroup(288,556)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊2D12
G = < a,b,c,d | a6=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a3b, dcd=c-1 >
Subgroups: 778 in 201 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C4⋊D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C4⋊Dic3, D6⋊C4, C6.D4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, D6⋊S3, S3×C12, C3×D12, C6×Dic3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, S3×C2×C6, C12⋊7D4, D6⋊3D4, D6⋊Dic3, C12⋊Dic3, C2×D6⋊S3, S3×C2×C12, C6×D12, D6⋊2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, S32, C2×D12, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, D6⋊S3, C2×S32, C12⋊7D4, D6⋊3D4, D12⋊5S3, S3×D12, C2×D6⋊S3, D6⋊2D12
►On 96 points
(1 79 5 83 9 75)(2 80 6 84 10 76)(3 81 7 73 11 77)(4 82 8 74 12 78)(13 47 17 39 21 43)(14 48 18 40 22 44)(15 37 19 41 23 45)(16 38 20 42 24 46)(25 69 33 65 29 61)(26 70 34 66 30 62)(27 71 35 67 31 63)(28 72 36 68 32 64)(49 96 57 92 53 88)(50 85 58 93 54 89)(51 86 59 94 55 90)(52 87 60 95 56 91)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 92)(14 93)(15 94)(16 95)(17 96)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(37 59)(38 60)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(61 82)(62 83)(63 84)(64 73)(65 74)(66 75)(67 76)(68 77)(69 78)(70 79)(71 80)(72 81)
(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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 48)(36 47)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 84)(59 83)(60 82)
G:=sub<Sym(96)| (1,79,5,83,9,75)(2,80,6,84,10,76)(3,81,7,73,11,77)(4,82,8,74,12,78)(13,47,17,39,21,43)(14,48,18,40,22,44)(15,37,19,41,23,45)(16,38,20,42,24,46)(25,69,33,65,29,61)(26,70,34,66,30,62)(27,71,35,67,31,63)(28,72,36,68,32,64)(49,96,57,92,53,88)(50,85,58,93,54,89)(51,86,59,94,55,90)(52,87,60,95,56,91), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,92)(14,93)(15,94)(16,95)(17,96)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(61,82)(62,83)(63,84)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81), (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,48)(36,47)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,84)(59,83)(60,82)>;
G:=Group( (1,79,5,83,9,75)(2,80,6,84,10,76)(3,81,7,73,11,77)(4,82,8,74,12,78)(13,47,17,39,21,43)(14,48,18,40,22,44)(15,37,19,41,23,45)(16,38,20,42,24,46)(25,69,33,65,29,61)(26,70,34,66,30,62)(27,71,35,67,31,63)(28,72,36,68,32,64)(49,96,57,92,53,88)(50,85,58,93,54,89)(51,86,59,94,55,90)(52,87,60,95,56,91), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,92)(14,93)(15,94)(16,95)(17,96)(18,85)(19,86)(20,87)(21,88)(22,89)(23,90)(24,91)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(61,82)(62,83)(63,84)(64,73)(65,74)(66,75)(67,76)(68,77)(69,78)(70,79)(71,80)(72,81), (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,72)(14,71)(15,70)(16,69)(17,68)(18,67)(19,66)(20,65)(21,64)(22,63)(23,62)(24,61)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,48)(36,47)(49,81)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,84)(59,83)(60,82) );
G=PermutationGroup([[(1,79,5,83,9,75),(2,80,6,84,10,76),(3,81,7,73,11,77),(4,82,8,74,12,78),(13,47,17,39,21,43),(14,48,18,40,22,44),(15,37,19,41,23,45),(16,38,20,42,24,46),(25,69,33,65,29,61),(26,70,34,66,30,62),(27,71,35,67,31,63),(28,72,36,68,32,64),(49,96,57,92,53,88),(50,85,58,93,54,89),(51,86,59,94,55,90),(52,87,60,95,56,91)], [(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,92),(14,93),(15,94),(16,95),(17,96),(18,85),(19,86),(20,87),(21,88),(22,89),(23,90),(24,91),(37,59),(38,60),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(61,82),(62,83),(63,84),(64,73),(65,74),(66,75),(67,76),(68,77),(69,78),(70,79),(71,80),(72,81)], [(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)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,96),(8,95),(9,94),(10,93),(11,92),(12,91),(13,72),(14,71),(15,70),(16,69),(17,68),(18,67),(19,66),(20,65),(21,64),(22,63),(23,62),(24,61),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,48),(36,47),(49,81),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,84),(59,83),(60,82)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 6P | 6Q | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | C4○D12 | S32 | S3×D4 | D4⋊2S3 | D6⋊S3 | C2×S32 | D12⋊5S3 | S3×D12 |
| kernel | D6⋊2D12 | D6⋊Dic3 | C12⋊Dic3 | C2×D6⋊S3 | S3×C2×C12 | C6×D12 | S3×C2×C4 | C2×D12 | C3×C12 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | C12 | D6 | C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 3 | 2 | 8 | 4 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of D6⋊2D12 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 5 | 5 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 7 | 2 | 0 | 0 | 0 | 0 |
| 2 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,0,10],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,3,5],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,2,0,0,0,0,2,6,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;
D6⋊2D12 in GAP, Magma, Sage, TeX
D_6\rtimes_2D_{12} % in TeX
G:=Group("D6:2D12"); // GroupNames label
G:=SmallGroup(288,556);
// by ID
G=gap.SmallGroup(288,556);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^12=d^2=1,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^-1>;
// generators/relations