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## G = C6.(C4⋊D4)  order 192 = 26·3

### 7th non-split extension by C6 of C4⋊D4 acting via C4⋊D4/C22⋊C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C6.(C4⋊D4)
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — C6.(C4⋊D4)
 Lower central C3 — C22×C6 — C6.(C4⋊D4)
 Upper central C1 — C23 — C2.C42

Generators and relations for C6.(C4⋊D4)
G = < a,b,c,d | a6=b4=c4=1, d2=a3, bab-1=cac-1=a-1, ad=da, cbc-1=a3b-1, dbd-1=b-1, dcd-1=a3c-1 >

Subgroups: 528 in 170 conjugacy classes, 55 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, Dic3⋊C4, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C23.11D4, C6.C42, C3×C2.C42, C2×Dic3⋊C4, C2×D6⋊C4, C6.(C4⋊D4)
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C4⋊D4, C22.D4, C4.4D4, C422C2, C4○D12, S3×D4, D42S3, Q83S3, C23.11D4, C423S3, C23.9D6, Dic3⋊D4, C23.11D6, D6.D4, C4⋊C4⋊S3, C6.(C4⋊D4)

Smallest permutation representation of C6.(C4⋊D4)
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 89 29 18)(2 88 30 17)(3 87 25 16)(4 86 26 15)(5 85 27 14)(6 90 28 13)(7 80 20 93)(8 79 21 92)(9 84 22 91)(10 83 23 96)(11 82 24 95)(12 81 19 94)(31 59 44 72)(32 58 45 71)(33 57 46 70)(34 56 47 69)(35 55 48 68)(36 60 43 67)(37 63 52 78)(38 62 53 77)(39 61 54 76)(40 66 49 75)(41 65 50 74)(42 64 51 73)
(1 67 9 74)(2 72 10 73)(3 71 11 78)(4 70 12 77)(5 69 7 76)(6 68 8 75)(13 32 92 37)(14 31 93 42)(15 36 94 41)(16 35 95 40)(17 34 96 39)(18 33 91 38)(19 62 26 57)(20 61 27 56)(21 66 28 55)(22 65 29 60)(23 64 30 59)(24 63 25 58)(43 81 50 86)(44 80 51 85)(45 79 52 90)(46 84 53 89)(47 83 54 88)(48 82 49 87)
(1 77 4 74)(2 78 5 75)(3 73 6 76)(7 68 10 71)(8 69 11 72)(9 70 12 67)(13 39 16 42)(14 40 17 37)(15 41 18 38)(19 60 22 57)(20 55 23 58)(21 56 24 59)(25 64 28 61)(26 65 29 62)(27 66 30 63)(31 92 34 95)(32 93 35 96)(33 94 36 91)(43 84 46 81)(44 79 47 82)(45 80 48 83)(49 88 52 85)(50 89 53 86)(51 90 54 87)```

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A ··· 4F 4G ··· 4L 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 S3 D4 D4 D6 C4○D4 C4○D12 S3×D4 D4⋊2S3 Q8⋊3S3 kernel C6.(C4⋊D4) C6.C42 C3×C2.C42 C2×Dic3⋊C4 C2×D6⋊C4 C2.C42 C2×Dic3 C22×S3 C22×C4 C2×C6 C22 C22 C22 C22 # reps 1 2 1 1 3 1 2 2 3 10 12 2 1 1

Matrix representation of C6.(C4⋊D4) in GL6(𝔽13)

 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 0 1 0 0 0 0 12 1
,
 0 8 0 0 0 0 5 0 0 0 0 0 0 0 1 3 0 0 0 0 8 12 0 0 0 0 0 0 4 11 0 0 0 0 2 9
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 7 3 0 0 0 0 10 6
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 1 8 0 0 0 0 0 0 10 6 0 0 0 0 7 3

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,5,0,0,0,0,8,0,0,0,0,0,0,0,1,8,0,0,0,0,3,12,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,7,10,0,0,0,0,3,6],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,0,0,0,8,0,0,0,0,0,0,10,7,0,0,0,0,6,3] >;`

C6.(C4⋊D4) in GAP, Magma, Sage, TeX

`C_6.(C_4\rtimes D_4)`
`% in TeX`

`G:=Group("C6.(C4:D4)");`
`// GroupNames label`

`G:=SmallGroup(192,234);`
`// by ID`

`G=gap.SmallGroup(192,234);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,64,590,387,100,6278]);`
`// Polycyclic`

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

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