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## G = D12.11D6order 288 = 25·32

### 11st non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.11D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×Dic6 — D12.11D6
 Lower central C32 — C3×C6 — C3×C12 — D12.11D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.11D6
G = < a,b,c,d | a12=b2=c6=1, d2=a6, bab=dad-1=a-1, cac-1=a5, cbc-1=a10b, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 466 in 129 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), SD16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C8⋊S3, C24⋊C2, C4.Dic3, D4.S3, Q82S3, C3⋊Q16, C3⋊Q16, C3×Q16, C2×Dic6, S3×Q8, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, S3×Dic3, C322Q8, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, C324Q8, Q8×C32, Q16⋊S3, Q8.14D6, D6.Dic3, Dic6⋊S3, D12.S3, C3×C3⋊Q16, C327Q16, S3×Dic6, C3×Q83S3, D12.11D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C8.C22, S32, S3×D4, C2×C3⋊D4, C2×S32, Q16⋊S3, Q8.14D6, S3×C3⋊D4, D12.11D6

Smallest permutation representation of D12.11D6
On 96 points
Generators in S96
```(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 65)(2 64)(3 63)(4 62)(5 61)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 59)(14 58)(15 57)(16 56)(17 55)(18 54)(19 53)(20 52)(21 51)(22 50)(23 49)(24 60)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 77)(38 76)(39 75)(40 74)(41 73)(42 84)(43 83)(44 82)(45 81)(46 80)(47 79)(48 78)
(1 49 5 57 9 53)(2 54 6 50 10 58)(3 59 7 55 11 51)(4 52 8 60 12 56)(13 61 17 69 21 65)(14 66 18 62 22 70)(15 71 19 67 23 63)(16 64 20 72 24 68)(25 83 33 75 29 79)(26 76 34 80 30 84)(27 81 35 73 31 77)(28 74 36 78 32 82)(37 88 45 92 41 96)(38 93 46 85 42 89)(39 86 47 90 43 94)(40 91 48 95 44 87)
(1 76 7 82)(2 75 8 81)(3 74 9 80)(4 73 10 79)(5 84 11 78)(6 83 12 77)(13 90 19 96)(14 89 20 95)(15 88 21 94)(16 87 22 93)(17 86 23 92)(18 85 24 91)(25 56 31 50)(26 55 32 49)(27 54 33 60)(28 53 34 59)(29 52 35 58)(30 51 36 57)(37 65 43 71)(38 64 44 70)(39 63 45 69)(40 62 46 68)(41 61 47 67)(42 72 48 66)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 24A 24B order 1 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 8 8 12 12 12 12 12 12 12 12 12 12 12 24 24 size 1 1 6 12 2 2 4 2 4 6 12 36 2 2 4 12 12 12 12 36 4 4 4 4 6 6 8 8 8 8 24 12 12

33 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 4 4 4 8 type + + + + + + + + + + + + + + + + + - + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8.C22 S32 S3×D4 C2×S32 Q16⋊S3 Q8.14D6 S3×C3⋊D4 D12.11D6 kernel D12.11D6 D6.Dic3 Dic6⋊S3 D12.S3 C3×C3⋊Q16 C32⋊7Q16 S3×Dic6 C3×Q8⋊3S3 C3⋊Q16 Q8⋊3S3 C3×Dic3 S3×C6 C3⋊C8 Dic6 C4×S3 D12 C3×Q8 Dic3 D6 C32 Q8 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 2 2 2 1

Matrix representation of D12.11D6 in GL8(𝔽73)

 72 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 1 72 0 0 0 0 0 0 0 0 43 13 0 0 0 0 0 0 60 30 0 0 0 0 0 0 0 0 15 47 0 7 0 0 0 0 47 58 66 0 0 0 0 0 0 66 15 47 0 0 0 0 7 0 47 58
,
 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 0 7 0 47 58 0 0 0 0 0 7 58 26 0 0 0 0 47 58 66 0 0 0 0 0 58 26 0 66
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 2 56 56 0 0 0 0 2 72 17 56 0 0 0 0 56 17 2 72 0 0 0 0 56 56 72 71

`G:=sub<GL(8,GF(73))| [72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,43,60,0,0,0,0,0,0,13,30,0,0,0,0,0,0,0,0,15,47,0,7,0,0,0,0,47,58,66,0,0,0,0,0,0,66,15,47,0,0,0,0,7,0,47,58],[0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,7,0,47,58,0,0,0,0,0,7,58,26,0,0,0,0,47,58,66,0,0,0,0,0,58,26,0,66],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,0,0,1,2,56,56,0,0,0,0,2,72,17,56,0,0,0,0,56,17,2,72,0,0,0,0,56,56,72,71] >;`

D12.11D6 in GAP, Magma, Sage, TeX

`D_{12}._{11}D_6`
`% in TeX`

`G:=Group("D12.11D6");`
`// GroupNames label`

`G:=SmallGroup(288,591);`
`// by ID`

`G=gap.SmallGroup(288,591);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,100,346,185,80,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^6,b*a*b=d*a*d^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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