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## G = D12.12D6order 288 = 25·32

### 12nd non-split extension by D12 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — D12.12D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D12.12D6
 Lower central C32 — C3×C6 — C3×C12 — D12.12D6
 Upper central C1 — C2 — C4 — Q8

Generators and relations for D12.12D6
G = < a,b,c,d | a12=b2=c6=1, d2=a6, bab=dad-1=a-1, cac-1=a5, cbc-1=a10b, dbd-1=ab, dcd-1=a6c-1 >

Subgroups: 498 in 133 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, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C6, S3×C8, C24⋊C2, C2×C3⋊C8, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C3×SD16, C4○D12, D42S3, Q83S3, C3×C4○D4, C3×C3⋊C8, C324C8, S3×Dic3, D6⋊S3, S3×C12, S3×C12, C3×D12, C3×D12, C324Q8, Q8×C32, Q8.7D6, Q8.13D6, S3×C3⋊C8, C322D8, D12.S3, C3×Q82S3, C327Q16, D125S3, C3×Q83S3, D12.12D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4○D8, S32, S3×D4, C2×C3⋊D4, C2×S32, Q8.7D6, Q8.13D6, S3×C3⋊D4, D12.12D6

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

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 6F 6G 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 24A 24B order 1 2 2 2 2 3 3 3 4 4 4 4 4 6 6 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 12 12 24 24 size 1 1 6 12 12 2 2 4 2 3 3 4 36 2 2 4 12 12 12 24 6 6 18 18 4 4 4 4 6 6 8 8 8 8 12 12

36 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 8 type + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C4○D8 S32 S3×D4 C2×S32 Q8.7D6 Q8.13D6 S3×C3⋊D4 D12.12D6 kernel D12.12D6 S3×C3⋊C8 C32⋊2D8 D12.S3 C3×Q8⋊2S3 C32⋊7Q16 D12⋊5S3 C3×Q8⋊3S3 Q8⋊2S3 Q8⋊3S3 C3×Dic3 S3×C6 C3⋊C8 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 2 2 2 2 4 1 1 1 2 2 2 1

Matrix representation of D12.12D6 in GL6(𝔽73)

 1 48 0 0 0 0 3 72 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 52 16 0 0 0 0 9 21 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 27 55 0 0 0 0 8 46 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 37 62 0 0 0 0 25 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 72 1

`G:=sub<GL(6,GF(73))| [1,3,0,0,0,0,48,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,9,0,0,0,0,16,21,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[27,8,0,0,0,0,55,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[37,25,0,0,0,0,62,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;`

D12.12D6 in GAP, Magma, Sage, TeX

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

`G:=Group("D12.12D6");`
`// GroupNames label`

`G:=SmallGroup(288,595);`
`// by ID`

`G=gap.SmallGroup(288,595);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,135,100,346,185,80,1356,9414]);`
`// 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*b,d*c*d^-1=a^6*c^-1>;`
`// generators/relations`

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