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## G = C42.188D6order 192 = 26·3

### 8th non-split extension by C42 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42.188D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C42.188D6
 Lower central C3 — C6 — C42.188D6
 Upper central C1 — C2×C4 — C42⋊C2

Generators and relations for C42.188D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 648 in 310 conjugacy classes, 155 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C4×C4○D4, S3×C42, C422S3, Dic34D4, Dic6⋊C4, Dic35D4, C2×C4×Dic3, C3×C42⋊C2, C2×C4○D12, C42.188D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C23×C4, C2×C4○D4, S3×C2×C4, S3×C23, C4×C4○D4, S3×C22×C4, S3×C4○D4, C42.188D6

Smallest permutation representation of C42.188D6
On 96 points
Generators in S96
```(1 85 37 83)(2 86 38 84)(3 87 39 79)(4 88 40 80)(5 89 41 81)(6 90 42 82)(7 28 54 55)(8 29 49 56)(9 30 50 57)(10 25 51 58)(11 26 52 59)(12 27 53 60)(13 69 21 61)(14 70 22 62)(15 71 23 63)(16 72 24 64)(17 67 19 65)(18 68 20 66)(31 91 43 74)(32 92 44 75)(33 93 45 76)(34 94 46 77)(35 95 47 78)(36 96 48 73)
(1 12 15 31)(2 54 16 44)(3 8 17 33)(4 50 18 46)(5 10 13 35)(6 52 14 48)(7 24 32 38)(9 20 34 40)(11 22 36 42)(19 45 39 49)(21 47 41 51)(23 43 37 53)(25 69 95 89)(26 62 96 82)(27 71 91 85)(28 64 92 84)(29 67 93 87)(30 66 94 80)(55 72 75 86)(56 65 76 79)(57 68 77 88)(58 61 78 81)(59 70 73 90)(60 63 74 83)
(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 65 37 67)(2 72 38 64)(3 63 39 71)(4 70 40 62)(5 61 41 69)(6 68 42 66)(7 75 54 92)(8 91 49 74)(9 73 50 96)(10 95 51 78)(11 77 52 94)(12 93 53 76)(13 81 21 89)(14 88 22 80)(15 79 23 87)(16 86 24 84)(17 83 19 85)(18 90 20 82)(25 47 58 35)(26 34 59 46)(27 45 60 33)(28 32 55 44)(29 43 56 31)(30 36 57 48)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E ··· 4N 4O ··· 4V 4W ··· 4AD 6A 6B 6C 6D 6E 12A 12B 12C 12D 12E ··· 12N order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 6 6 6 6 2 1 1 1 1 2 ··· 2 3 ··· 3 6 ··· 6 2 2 2 4 4 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 D6 C4○D4 C4×S3 S3×C4○D4 kernel C42.188D6 S3×C42 C42⋊2S3 Dic3⋊4D4 Dic6⋊C4 Dic3⋊5D4 C2×C4×Dic3 C3×C42⋊C2 C2×C4○D12 C4○D12 C42⋊C2 C42 C22⋊C4 C4⋊C4 C22×C4 Dic3 C2×C4 C2 # reps 1 2 2 4 2 2 1 1 1 16 1 2 2 2 1 8 8 4

Matrix representation of C42.188D6 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 8 0 0 0 0 8 0 0 0 0 0 1 0 0 12 0
,
 0 1 0 0 12 1 0 0 0 0 0 12 0 0 12 0
,
 1 0 0 0 1 12 0 0 0 0 8 0 0 0 0 5
`G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,8,0,0,0,0,0,12,0,0,1,0],[0,12,0,0,1,1,0,0,0,0,0,12,0,0,12,0],[1,1,0,0,0,12,0,0,0,0,8,0,0,0,0,5] >;`

C42.188D6 in GAP, Magma, Sage, TeX

`C_4^2._{188}D_6`
`% in TeX`

`G:=Group("C4^2.188D6");`
`// GroupNames label`

`G:=SmallGroup(192,1081);`
`// by ID`

`G=gap.SmallGroup(192,1081);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,297,80,6278]);`
`// Polycyclic`

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

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