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## G = C4⋊C4.234D6order 192 = 26·3

### 12nd non-split extension by C4⋊C4 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4⋊C4.234D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C4⋊C4.234D6
 Lower central C3 — C6 — C12 — C4⋊C4.234D6
 Upper central C1 — C2×C4 — C22×C4 — C42⋊C2

Generators and relations for C4⋊C4.234D6
G = < a,b,c,d | a4=b4=c6=1, d2=ab2, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b-1, dcd-1=c-1 >

Subgroups: 232 in 114 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×6], C22×C4, C3⋊C8 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×C6, C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C42⋊C2, C22×C8, C2×C3⋊C8 [×2], C2×C3⋊C8 [×4], C4×Dic3, C4⋊Dic3 [×2], C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C22×C12, C23.25D4, C6.Q16 [×2], C12.Q8 [×2], C22×C3⋊C8, C23.26D6, C3×C42⋊C2, C4⋊C4.234D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C4○D8 [×2], Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, C23.25D4, C2×Dic3⋊C4, Q8.13D6 [×2], C4⋊C4.234D6

Smallest permutation representation of C4⋊C4.234D6
On 96 points
Generators in S96
```(1 86 5 82)(2 87 6 83)(3 88 7 84)(4 81 8 85)(9 39 13 35)(10 40 14 36)(11 33 15 37)(12 34 16 38)(17 96 21 92)(18 89 22 93)(19 90 23 94)(20 91 24 95)(25 58 29 62)(26 59 30 63)(27 60 31 64)(28 61 32 57)(41 71 45 67)(42 72 46 68)(43 65 47 69)(44 66 48 70)(49 75 53 79)(50 76 54 80)(51 77 55 73)(52 78 56 74)
(1 76 84 52)(2 79 85 55)(3 74 86 50)(4 77 87 53)(5 80 88 56)(6 75 81 51)(7 78 82 54)(8 73 83 49)(9 47 37 67)(10 42 38 70)(11 45 39 65)(12 48 40 68)(13 43 33 71)(14 46 34 66)(15 41 35 69)(16 44 36 72)(17 59 94 28)(18 62 95 31)(19 57 96 26)(20 60 89 29)(21 63 90 32)(22 58 91 27)(23 61 92 30)(24 64 93 25)
(1 22 65)(2 66 23)(3 24 67)(4 68 17)(5 18 69)(6 70 19)(7 20 71)(8 72 21)(9 78 64 13 74 60)(10 61 75 14 57 79)(11 80 58 15 76 62)(12 63 77 16 59 73)(25 33 50 29 37 54)(26 55 38 30 51 34)(27 35 52 31 39 56)(28 49 40 32 53 36)(41 88 95)(42 96 81)(43 82 89)(44 90 83)(45 84 91)(46 92 85)(47 86 93)(48 94 87)
(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)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A 6B 6C 6D 6E 8A ··· 8H 12A 12B 12C 12D 12E ··· 12N order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 1 1 1 1 2 2 4 4 4 4 12 12 12 12 2 2 2 4 4 6 ··· 6 2 2 2 2 4 ··· 4

48 irreducible representations

 dim 1 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 C4 S3 D4 Q8 D4 D6 D6 Dic6 C4×S3 C3⋊D4 C3⋊D4 C4○D8 Q8.13D6 kernel C4⋊C4.234D6 C6.Q16 C12.Q8 C22×C3⋊C8 C23.26D6 C3×C42⋊C2 C2×C3⋊C8 C42⋊C2 C2×C12 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 2 2 1 1 1 8 1 1 2 1 2 1 4 4 2 2 8 4

Matrix representation of C4⋊C4.234D6 in GL6(𝔽73)

 27 0 0 0 0 0 55 46 0 0 0 0 0 0 46 0 0 0 0 0 25 27 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 72 70 0 0 0 0 25 1 0 0 0 0 0 0 27 70 0 0 0 0 48 46 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 48 72 0 0 0 0 0 0 1 0 0 0 0 0 18 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 22 0 0 0 0 0 38 63 0 0 0 0 0 0 51 0 0 0 0 0 4 10 0 0 0 0 0 0 1 0 0 0 0 0 1 72

`G:=sub<GL(6,GF(73))| [27,55,0,0,0,0,0,46,0,0,0,0,0,0,46,25,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,25,0,0,0,0,70,1,0,0,0,0,0,0,27,48,0,0,0,0,70,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,48,0,0,0,0,0,72,0,0,0,0,0,0,1,18,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[22,38,0,0,0,0,0,63,0,0,0,0,0,0,51,4,0,0,0,0,0,10,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;`

C4⋊C4.234D6 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{234}D_6`
`% in TeX`

`G:=Group("C4:C4.234D6");`
`// GroupNames label`

`G:=SmallGroup(192,557);`
`// by ID`

`G=gap.SmallGroup(192,557);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,387,58,438,102,6278]);`
`// Polycyclic`

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

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