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## G = C4⋊C4.232D6order 192 = 26·3

### 10th 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⋊C4.232D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — C4⋊C4.232D6
 Lower central C3 — C6 — C12 — C4⋊C4.232D6
 Upper central C1 — C22 — C22×C4 — C42⋊C2

Generators and relations for C4⋊C4.232D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab-1, dcd-1=c-1 >

Subgroups: 264 in 118 conjugacy classes, 63 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×6], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4, C3⋊C8 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×C6, C4.Q8 [×2], C2.D8 [×2], C2×C4⋊C4, C42⋊C2, C2×M4(2), C2×C3⋊C8 [×2], C4.Dic3 [×4], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C22×Dic3, C22×C12, M4(2)⋊C4, C6.Q16 [×2], C12.Q8 [×2], C2×C4.Dic3, C2×C4⋊Dic3, C3×C42⋊C2, C4⋊C4.232D6
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, C8⋊C22, C8.C22, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, M4(2)⋊C4, C2×Dic3⋊C4, D4⋊D6, Q8.14D6, C4⋊C4.232D6

Smallest permutation representation of C4⋊C4.232D6
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 40 45 38)(26 41 43 39)(27 42 44 37)(28 32 47 34)(29 33 48 35)(30 31 46 36)(49 55 52 58)(50 56 53 59)(51 57 54 60)(61 93 64 96)(62 94 65 91)(63 95 66 92)(67 75 70 78)(68 76 71 73)(69 77 72 74)(79 86 82 89)(80 87 83 90)(81 88 84 85)
(1 49 7 74)(2 53 8 78)(3 51 9 76)(4 52 10 77)(5 50 11 75)(6 54 12 73)(13 69 19 55)(14 67 20 59)(15 71 21 57)(16 72 22 58)(17 70 23 56)(18 68 24 60)(25 84 47 65)(26 82 48 63)(27 80 46 61)(28 62 45 81)(29 66 43 79)(30 64 44 83)(31 93 37 87)(32 91 38 85)(33 95 39 89)(34 94 40 88)(35 92 41 86)(36 96 42 90)
(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 38 7 32)(2 37 8 31)(3 39 9 33)(4 40 10 34)(5 42 11 36)(6 41 12 35)(13 47 19 25)(14 46 20 27)(15 48 21 26)(16 28 22 45)(17 30 23 44)(18 29 24 43)(49 65 74 84)(50 64 75 83)(51 63 76 82)(52 62 77 81)(53 61 78 80)(54 66 73 79)(55 94 69 88)(56 93 70 87)(57 92 71 86)(58 91 72 85)(59 96 67 90)(60 95 68 89)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + - + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Q8 D4 D6 D6 Dic6 C4×S3 C3⋊D4 C3⋊D4 C8⋊C22 C8.C22 D4⋊D6 Q8.14D6 kernel C4⋊C4.232D6 C6.Q16 C12.Q8 C2×C4.Dic3 C2×C4⋊Dic3 C3×C42⋊C2 C4.Dic3 C42⋊C2 C2×C12 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C6 C2 C2 # reps 1 2 2 1 1 1 8 1 1 2 1 2 1 4 4 2 2 1 1 2 2

Matrix representation of C4⋊C4.232D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 66 59 0 0 0 0 14 7 0 0 0 0 0 0 7 14 0 0 0 0 59 66
,
 7 59 0 0 0 0 14 66 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 1 0 0 0 0 72 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 72 0
,
 25 37 0 0 0 0 62 48 0 0 0 0 0 0 54 29 0 0 0 0 48 19 0 0 0 0 0 0 45 42 0 0 0 0 70 28

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,14,0,0,0,0,59,7,0,0,0,0,0,0,7,59,0,0,0,0,14,66],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[25,62,0,0,0,0,37,48,0,0,0,0,0,0,54,48,0,0,0,0,29,19,0,0,0,0,0,0,45,70,0,0,0,0,42,28] >;`

C4⋊C4.232D6 in GAP, Magma, Sage, TeX

`C_4\rtimes C_4._{232}D_6`
`% in TeX`

`G:=Group("C4:C4.232D6");`
`// GroupNames label`

`G:=SmallGroup(192,554);`
`// by ID`

`G=gap.SmallGroup(192,554);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,254,387,100,1123,297,136,6278]);`
`// Polycyclic`

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

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