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## G = C4⋊C4.228D6order 192 = 26·3

### 6th non-split extension by C4⋊C4 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C4⋊C4.228D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C12⋊7D4 — C4⋊C4.228D6
 Lower central C3 — C6 — C2×C12 — C4⋊C4.228D6
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for C4⋊C4.228D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-1 >

Subgroups: 344 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], S3, C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, Dic3, C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×2], C3⋊C8 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C22⋊C8, D4⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C4⋊D4, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C23.46D4, C12.Q8 [×2], C6.D8 [×2], C12.55D4, C127D4, C6×C4⋊C4, C4⋊C4.228D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C2×SD16, C8⋊C22, Q82S3 [×2], C4○D12 [×2], C2×C3⋊D4, C23.46D4, C23.28D6, D126C22, C2×Q82S3, C4⋊C4.228D6

Smallest permutation representation of C4⋊C4.228D6
On 96 points
Generators in S96
```(1 21 73 41)(2 22 74 42)(3 23 75 37)(4 24 76 38)(5 19 77 39)(6 20 78 40)(7 72 52 86)(8 67 53 87)(9 68 54 88)(10 69 49 89)(11 70 50 90)(12 71 51 85)(13 63 33 83)(14 64 34 84)(15 65 35 79)(16 66 36 80)(17 61 31 81)(18 62 32 82)(25 95 45 59)(26 96 46 60)(27 91 47 55)(28 92 48 56)(29 93 43 57)(30 94 44 58)
(1 44 35 68)(2 45 36 69)(3 46 31 70)(4 47 32 71)(5 48 33 72)(6 43 34 67)(7 19 92 83)(8 20 93 84)(9 21 94 79)(10 22 95 80)(11 23 96 81)(12 24 91 82)(13 86 77 28)(14 87 78 29)(15 88 73 30)(16 89 74 25)(17 90 75 26)(18 85 76 27)(37 60 61 50)(38 55 62 51)(39 56 63 52)(40 57 64 53)(41 58 65 54)(42 59 66 49)
(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 6 15 14)(2 13 16 5)(3 4 17 18)(7 69 56 25)(8 30 57 68)(9 67 58 29)(10 28 59 72)(11 71 60 27)(12 26 55 70)(19 42 63 80)(20 79 64 41)(21 40 65 84)(22 83 66 39)(23 38 61 82)(24 81 62 37)(31 32 75 76)(33 36 77 74)(34 73 78 35)(43 54 87 94)(44 93 88 53)(45 52 89 92)(46 91 90 51)(47 50 85 96)(48 95 86 49)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C ··· 4G 4H 6A ··· 6G 8A 8B 8C 8D 12A ··· 12L order 1 2 2 2 2 2 2 3 4 4 4 ··· 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 24 2 2 2 4 ··· 4 24 2 ··· 2 12 12 12 12 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 SD16 C3⋊D4 C3⋊D4 C4○D12 C8⋊C22 Q8⋊2S3 D12⋊6C22 kernel C4⋊C4.228D6 C12.Q8 C6.D8 C12.55D4 C12⋊7D4 C6×C4⋊C4 C2×C4⋊C4 C2×C12 C22×C6 C4⋊C4 C22×C4 C12 C2×C6 C2×C4 C23 C4 C6 C22 C2 # reps 1 2 2 1 1 1 1 1 1 2 1 4 4 2 2 8 1 2 2

Matrix representation of C4⋊C4.228D6 in GL6(𝔽73)

 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 72 2 0 0 0 0 72 1
,
 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 61 12 0 0 0 0 67 12
,
 1 1 0 0 0 0 72 0 0 0 0 0 0 0 0 27 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 1 0 0 0 0 0 72 0 0 0 0 0 0 0 46 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,61,67,0,0,0,0,12,12],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;`

C4⋊C4.228D6 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{228}D_6`
`% in TeX`

`G:=Group("C4:C4.228D6");`
`// GroupNames label`

`G:=SmallGroup(192,527);`
`// by ID`

`G=gap.SmallGroup(192,527);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,268,1123,297,136,6278]);`
`// Polycyclic`

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

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