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## G = C23⋊3D12order 192 = 26·3

### 1st semidirect product of C23 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C23⋊3D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×D12 — C23⋊3D12
 Lower central C3 — C22×C6 — C23⋊3D12
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C233D12
G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, dad-1=ac=ca, eae=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1080 in 322 conjugacy classes, 67 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C3, C4 [×7], C22 [×3], C22 [×4], C22 [×30], S3 [×4], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×2], C2×C4 [×13], D4 [×24], C23, C23 [×2], C23 [×22], Dic3 [×4], C12 [×3], D6 [×20], C2×C6 [×3], C2×C6 [×4], C2×C6 [×10], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×2], C2×D4 [×18], C24, C24 [×2], D12 [×8], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×2], C2×C12 [×5], C22×S3 [×4], C22×S3 [×12], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C22×D4 [×3], D6⋊C4 [×4], C3×C22⋊C4 [×2], C2×D12 [×6], C22×Dic3 [×2], C2×C3⋊D4 [×12], C22×C12 [×2], S3×C23 [×2], C23×C6, C232D4, C6.C42, C2×D6⋊C4 [×2], C6×C22⋊C4, C22×D12, C22×C3⋊D4 [×2], C233D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×12], C23, D6 [×3], C2×D4 [×6], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×D12, C4○D12, S3×D4 [×4], C2×C3⋊D4, C232D4, D6⋊D4 [×2], Dic3⋊D4 [×2], C127D4, C232D6, C123D4, C233D12

Smallest permutation representation of C233D12
On 96 points
Generators in S96
```(1 82)(2 72)(3 84)(4 62)(5 74)(6 64)(7 76)(8 66)(9 78)(10 68)(11 80)(12 70)(13 25)(14 47)(15 27)(16 37)(17 29)(18 39)(19 31)(20 41)(21 33)(22 43)(23 35)(24 45)(26 94)(28 96)(30 86)(32 88)(34 90)(36 92)(38 85)(40 87)(42 89)(44 91)(46 93)(48 95)(49 65)(50 77)(51 67)(52 79)(53 69)(54 81)(55 71)(56 83)(57 61)(58 73)(59 63)(60 75)
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 81)(14 82)(15 83)(16 84)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 49)(33 50)(34 51)(35 52)(36 53)(61 96)(62 85)(63 86)(64 87)(65 88)(66 89)(67 90)(68 91)(69 92)(70 93)(71 94)(72 95)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 93)(14 94)(15 95)(16 96)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(61 84)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 81)(71 82)(72 83)
(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 82)(14 81)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(21 74)(22 73)(23 84)(24 83)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 96)(69 95)(70 94)(71 93)(72 92)```

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 12 12 12 12 2 4 4 4 4 12 12 12 12 2 ··· 2 4 4 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 C4○D4 C3⋊D4 D12 C4○D12 S3×D4 kernel C23⋊3D12 C6.C42 C2×D6⋊C4 C6×C22⋊C4 C22×D12 C22×C3⋊D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×S3 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 # reps 1 1 2 1 1 2 1 4 2 4 2 2 1 2 4 4 4 4

Matrix representation of C233D12 in GL6(𝔽13)

 1 11 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 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 12 0 0 0 0 0 0 12
,
 8 10 0 0 0 0 0 5 0 0 0 0 0 0 3 10 0 0 0 0 3 6 0 0 0 0 0 0 1 0 0 0 0 0 0 12
,
 8 10 0 0 0 0 8 5 0 0 0 0 0 0 3 10 0 0 0 0 7 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,11,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,10,5,0,0,0,0,0,0,3,3,0,0,0,0,10,6,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[8,8,0,0,0,0,10,5,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C233D12 in GAP, Magma, Sage, TeX

`C_2^3\rtimes_3D_{12}`
`% in TeX`

`G:=Group("C2^3:3D12");`
`// GroupNames label`

`G:=SmallGroup(192,519);`
`// by ID`

`G=gap.SmallGroup(192,519);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,6278]);`
`// Polycyclic`

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

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