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## G = D12.37D4order 192 = 26·3

### 7th non-split extension by D12 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — D12.37D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C2×C4○D12 — D12.37D4
 Lower central C3 — C6 — C2×C12 — D12.37D4
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for D12.37D4
G = < a,b,c,d | a12=b2=c4=1, d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >

Subgroups: 448 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×5], C22, C22 [×7], S3 [×2], C6 [×3], C6, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×7], Q8 [×5], C23, C23, Dic3 [×2], C12 [×2], C12 [×3], D6 [×4], C2×C6, C2×C6 [×3], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C2×Q8, C4○D4 [×4], C3⋊C8 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3, C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×S3, C22×C6, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C2×C3⋊C8 [×2], Q82S3 [×2], C3⋊Q16 [×2], C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C2×C3⋊D4, C22×C12, C6×Q8, D4.7D4, C6.D8, C6.SD16, C12.55D4, C2×Q82S3, C2×C3⋊Q16, C3×C22⋊Q8, C2×C4○D12, D12.37D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C4○D8, C8.C22, S3×D4 [×2], C2×C3⋊D4, D4.7D4, C232D6, Q8.11D6, Q8.13D6, D12.37D4

Smallest permutation representation of D12.37D4
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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 18)(15 17)(20 24)(21 23)(25 27)(28 36)(29 35)(30 34)(31 33)(37 41)(38 40)(42 48)(43 47)(44 46)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 70)(62 69)(63 68)(64 67)(65 66)(71 72)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 24 66 47)(2 19 67 42)(3 14 68 37)(4 21 69 44)(5 16 70 39)(6 23 71 46)(7 18 72 41)(8 13 61 48)(9 20 62 43)(10 15 63 38)(11 22 64 45)(12 17 65 40)(25 58 91 73)(26 53 92 80)(27 60 93 75)(28 55 94 82)(29 50 95 77)(30 57 96 84)(31 52 85 79)(32 59 86 74)(33 54 87 81)(34 49 88 76)(35 56 89 83)(36 51 90 78)
(1 76 7 82)(2 77 8 83)(3 78 9 84)(4 79 10 73)(5 80 11 74)(6 81 12 75)(13 89 19 95)(14 90 20 96)(15 91 21 85)(16 92 22 86)(17 93 23 87)(18 94 24 88)(25 44 31 38)(26 45 32 39)(27 46 33 40)(28 47 34 41)(29 48 35 42)(30 37 36 43)(49 72 55 66)(50 61 56 67)(51 62 57 68)(52 63 58 69)(53 64 59 70)(54 65 60 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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,18)(15,17)(20,24)(21,23)(25,27)(28,36)(29,35)(30,34)(31,33)(37,41)(38,40)(42,48)(43,47)(44,46)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,70)(62,69)(63,68)(64,67)(65,66)(71,72)(73,78)(74,77)(75,76)(79,84)(80,83)(81,82)(85,87)(88,96)(89,95)(90,94)(91,93), (1,24,66,47)(2,19,67,42)(3,14,68,37)(4,21,69,44)(5,16,70,39)(6,23,71,46)(7,18,72,41)(8,13,61,48)(9,20,62,43)(10,15,63,38)(11,22,64,45)(12,17,65,40)(25,58,91,73)(26,53,92,80)(27,60,93,75)(28,55,94,82)(29,50,95,77)(30,57,96,84)(31,52,85,79)(32,59,86,74)(33,54,87,81)(34,49,88,76)(35,56,89,83)(36,51,90,78), (1,76,7,82)(2,77,8,83)(3,78,9,84)(4,79,10,73)(5,80,11,74)(6,81,12,75)(13,89,19,95)(14,90,20,96)(15,91,21,85)(16,92,22,86)(17,93,23,87)(18,94,24,88)(25,44,31,38)(26,45,32,39)(27,46,33,40)(28,47,34,41)(29,48,35,42)(30,37,36,43)(49,72,55,66)(50,61,56,67)(51,62,57,68)(52,63,58,69)(53,64,59,70)(54,65,60,71)>;`

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 12 12 size 1 1 1 1 4 12 12 2 2 2 2 2 8 8 12 12 2 2 2 4 4 12 12 12 12 4 4 4 4 8 8 8 8

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 type + + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 D6 C3⋊D4 C3⋊D4 C4○D8 C8.C22 S3×D4 Q8.11D6 Q8.13D6 kernel D12.37D4 C6.D8 C6.SD16 C12.55D4 C2×Q8⋊2S3 C2×C3⋊Q16 C3×C22⋊Q8 C2×C4○D12 C22⋊Q8 Dic6 D12 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×Q8 C2×C4 C23 C6 C6 C4 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 2 4 1 2 2 2

Matrix representation of D12.37D4 in GL6(𝔽73)

 1 1 0 0 0 0 72 0 0 0 0 0 0 0 1 71 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 1 0 0 0 0 0 72 0 0 0 0 0 0 1 71 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 56 1
,
 43 13 0 0 0 0 60 30 0 0 0 0 0 0 41 32 0 0 0 0 57 32 0 0 0 0 0 0 17 71 0 0 0 0 72 56
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 72 0 0 0 0 0 56 1

`G:=sub<GL(6,GF(73))| [1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,71,72,0,0,0,0,0,0,72,56,0,0,0,0,0,1],[43,60,0,0,0,0,13,30,0,0,0,0,0,0,41,57,0,0,0,0,32,32,0,0,0,0,0,0,17,72,0,0,0,0,71,56],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,56,0,0,0,0,0,1] >;`

D12.37D4 in GAP, Magma, Sage, TeX

`D_{12}._{37}D_4`
`% in TeX`

`G:=Group("D12.37D4");`
`// GroupNames label`

`G:=SmallGroup(192,606);`
`// by ID`

`G=gap.SmallGroup(192,606);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,184,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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