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## G = C4⋊C4.231D6order 192 = 26·3

### 9th 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⋊C4.231D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.48D4 — C4⋊C4.231D6
 Lower central C3 — C6 — C2×C12 — C4⋊C4.231D6
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for C4⋊C4.231D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=b2c-1 >

Subgroups: 248 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×6], C22×C6, C22⋊C8, Q8⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8 [×2], Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C22×C12, C22×C12, C23.47D4, C12.Q8 [×2], C6.SD16 [×2], C12.55D4, C12.48D4, C6×C4⋊C4, C4⋊C4.231D6
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, D4.S3 [×2], C4○D12 [×2], C2×C3⋊D4, C23.47D4, C23.28D6, C2×D4.S3, Q8.11D6, C4⋊C4.231D6

Smallest permutation representation of C4⋊C4.231D6
On 96 points
Generators in S96
```(1 22 4 19)(2 23 5 20)(3 24 6 21)(7 16 10 13)(8 17 11 14)(9 18 12 15)(25 43 28 46)(26 44 29 47)(27 45 30 48)(31 37 34 40)(32 38 35 41)(33 39 36 42)(49 69 64 59)(50 70 65 60)(51 71 66 55)(52 72 61 56)(53 67 62 57)(54 68 63 58)(73 96 85 83)(74 91 86 84)(75 92 87 79)(76 93 88 80)(77 94 89 81)(78 95 90 82)
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 43 19 37)(14 44 20 38)(15 45 21 39)(16 46 22 40)(17 47 23 41)(18 48 24 42)(49 96 52 93)(50 91 53 94)(51 92 54 95)(55 87 58 90)(56 88 59 85)(57 89 60 86)(61 80 64 83)(62 81 65 84)(63 82 66 79)(67 77 70 74)(68 78 71 75)(69 73 72 76)
(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 63 10 51)(2 65 11 53)(3 61 12 49)(4 54 7 66)(5 50 8 62)(6 52 9 64)(13 55 22 68)(14 57 23 70)(15 59 24 72)(16 71 19 58)(17 67 20 60)(18 69 21 56)(25 75 34 90)(26 77 35 86)(27 73 36 88)(28 87 31 78)(29 89 32 74)(30 85 33 76)(37 82 46 92)(38 84 47 94)(39 80 48 96)(40 95 43 79)(41 91 44 81)(42 93 45 83)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C ··· 4G 4H 4I 6A ··· 6G 8A 8B 8C 8D 12A ··· 12L order 1 2 2 2 2 2 3 4 4 4 ··· 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 2 2 2 4 ··· 4 24 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 D4.S3 Q8.11D6 kernel C4⋊C4.231D6 C12.Q8 C6.SD16 C12.55D4 C12.48D4 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⋊C4.231D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 27 0 0 0 0 0 28 46 0 0 0 0 0 0 1 0 0 0 0 0 26 72 0 0 0 0 0 0 56 71 0 0 0 0 71 17
,
 1 0 0 0 0 0 47 72 0 0 0 0 0 0 64 0 0 0 0 0 71 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 29 0 0 0 0 68 61 0 0 0 0 0 0 61 29 0 0 0 0 58 12 0 0 0 0 0 0 32 17 0 0 0 0 17 41

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[27,28,0,0,0,0,0,46,0,0,0,0,0,0,1,26,0,0,0,0,0,72,0,0,0,0,0,0,56,71,0,0,0,0,71,17],[1,47,0,0,0,0,0,72,0,0,0,0,0,0,64,71,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,68,0,0,0,0,29,61,0,0,0,0,0,0,61,58,0,0,0,0,29,12,0,0,0,0,0,0,32,17,0,0,0,0,17,41] >;`

C4⋊C4.231D6 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{231}D_6`
`% in TeX`

`G:=Group("C4:C4.231D6");`
`// GroupNames label`

`G:=SmallGroup(192,530);`
`// by ID`

`G=gap.SmallGroup(192,530);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,100,1123,297,136,6278]);`
`// 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^-1*b,d*c*d^-1=b^2*c^-1>;`
`// generators/relations`

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