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## G = D6⋊2D12order 288 = 25·32

### 2nd semidirect product of D6 and D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D6⋊2D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — D6⋊2D12
 Lower central C32 — C62 — D6⋊2D12
 Upper central C1 — C22 — C2×C4

Generators and relations for D62D12
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 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22 [×10], S3 [×4], C6 [×6], C6 [×7], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×7], C12 [×4], C12 [×3], D6 [×2], D6 [×8], C2×C6 [×2], C2×C6 [×11], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×4], C3×C6 [×3], C4×S3 [×2], D12 [×2], C2×Dic3, C2×Dic3 [×6], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C3×D4 [×2], C22×S3, C22×S3 [×2], C22×C6 [×3], C4⋊D4, C3×Dic3, C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×8], C62, C4⋊Dic3 [×3], D6⋊C4 [×2], C6.D4 [×2], S3×C2×C4, C2×D12, C2×C3⋊D4 [×4], C22×C12, C6×D4, D6⋊S3 [×4], S3×C12 [×2], C3×D12 [×2], C6×Dic3, C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, S3×C2×C6 [×2], C127D4, D63D4, D6⋊Dic3 [×2], C12⋊Dic3, C2×D6⋊S3 [×2], S3×C2×C12, C6×D12, D62D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×4], C22×S3 [×2], C4⋊D4, S32, C2×D12, C4○D12, S3×D4, D42S3, C2×C3⋊D4 [×2], D6⋊S3 [×2], C2×S32, C127D4, D63D4, D125S3, S3×D12, C2×D6⋊S3, D62D12

Smallest permutation representation of D62D12
On 96 points
Generators in S96
```(1 58 5 50 9 54)(2 59 6 51 10 55)(3 60 7 52 11 56)(4 49 8 53 12 57)(13 41 21 37 17 45)(14 42 22 38 18 46)(15 43 23 39 19 47)(16 44 24 40 20 48)(25 66 33 62 29 70)(26 67 34 63 30 71)(27 68 35 64 31 72)(28 69 36 65 32 61)(73 92 77 96 81 88)(74 93 78 85 82 89)(75 94 79 86 83 90)(76 95 80 87 84 91)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 73)(34 74)(35 75)(36 76)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 85)(72 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)
(1 25)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 84)(14 83)(15 82)(16 81)(17 80)(18 79)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(37 95)(38 94)(39 93)(40 92)(41 91)(42 90)(43 89)(44 88)(45 87)(46 86)(47 85)(48 96)(49 63)(50 62)(51 61)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)```

`G:=sub<Sym(96)| (1,58,5,50,9,54)(2,59,6,51,10,55)(3,60,7,52,11,56)(4,49,8,53,12,57)(13,41,21,37,17,45)(14,42,22,38,18,46)(15,43,23,39,19,47)(16,44,24,40,20,48)(25,66,33,62,29,70)(26,67,34,63,30,71)(27,68,35,64,31,72)(28,69,36,65,32,61)(73,92,77,96,81,88)(74,93,78,85,82,89)(75,94,79,86,83,90)(76,95,80,87,84,91), (1,44)(2,45)(3,46)(4,47)(5,48)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,85)(72,86), (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), (1,25)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,63)(50,62)(51,61)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)>;`

`G:=Group( (1,58,5,50,9,54)(2,59,6,51,10,55)(3,60,7,52,11,56)(4,49,8,53,12,57)(13,41,21,37,17,45)(14,42,22,38,18,46)(15,43,23,39,19,47)(16,44,24,40,20,48)(25,66,33,62,29,70)(26,67,34,63,30,71)(27,68,35,64,31,72)(28,69,36,65,32,61)(73,92,77,96,81,88)(74,93,78,85,82,89)(75,94,79,86,83,90)(76,95,80,87,84,91), (1,44)(2,45)(3,46)(4,47)(5,48)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,49)(20,50)(21,51)(22,52)(23,53)(24,54)(25,77)(26,78)(27,79)(28,80)(29,81)(30,82)(31,83)(32,84)(33,73)(34,74)(35,75)(36,76)(61,87)(62,88)(63,89)(64,90)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,85)(72,86), (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), (1,25)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(37,95)(38,94)(39,93)(40,92)(41,91)(42,90)(43,89)(44,88)(45,87)(46,86)(47,85)(48,96)(49,63)(50,62)(51,61)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64) );`

`G=PermutationGroup([(1,58,5,50,9,54),(2,59,6,51,10,55),(3,60,7,52,11,56),(4,49,8,53,12,57),(13,41,21,37,17,45),(14,42,22,38,18,46),(15,43,23,39,19,47),(16,44,24,40,20,48),(25,66,33,62,29,70),(26,67,34,63,30,71),(27,68,35,64,31,72),(28,69,36,65,32,61),(73,92,77,96,81,88),(74,93,78,85,82,89),(75,94,79,86,83,90),(76,95,80,87,84,91)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,49),(20,50),(21,51),(22,52),(23,53),(24,54),(25,77),(26,78),(27,79),(28,80),(29,81),(30,82),(31,83),(32,84),(33,73),(34,74),(35,75),(36,76),(61,87),(62,88),(63,89),(64,90),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,85),(72,86)], [(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)], [(1,25),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,84),(14,83),(15,82),(16,81),(17,80),(18,79),(19,78),(20,77),(21,76),(22,75),(23,74),(24,73),(37,95),(38,94),(39,93),(40,92),(41,91),(42,90),(43,89),(44,88),(45,87),(46,86),(47,85),(48,96),(49,63),(50,62),(51,61),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64)])`

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 D62D12 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] >;`

D62D12 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`

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