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## G = C42.125D6order 192 = 26·3

### 125th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42.125D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — C42.125D6
 Lower central C3 — C6 — C42.125D6
 Upper central C1 — C22 — C4×Q8

Generators and relations for C42.125D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c5 >

Subgroups: 504 in 266 conjugacy classes, 151 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C42⋊C2, C4×Q8, C4×Q8, C22×Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×Q8, C6×Q8, C23.32C23, C4×Dic6, C422S3, Dic6⋊C4, C4⋊C47S3, Q8×Dic3, Q8×C12, C2×S3×Q8, C42.125D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, 2- 1+4, S3×C2×C4, S3×C23, C23.32C23, S3×C22×C4, Q8.15D6, Q8○D12, C42.125D6

Smallest permutation representation of C42.125D6
On 96 points
Generators in S96
```(1 15 7 21)(2 22 8 16)(3 17 9 23)(4 24 10 18)(5 19 11 13)(6 14 12 20)(25 78 31 84)(26 73 32 79)(27 80 33 74)(28 75 34 81)(29 82 35 76)(30 77 36 83)(37 87 43 93)(38 94 44 88)(39 89 45 95)(40 96 46 90)(41 91 47 85)(42 86 48 92)(49 71 55 65)(50 66 56 72)(51 61 57 67)(52 68 58 62)(53 63 59 69)(54 70 60 64)
(1 42 57 82)(2 43 58 83)(3 44 59 84)(4 45 60 73)(5 46 49 74)(6 47 50 75)(7 48 51 76)(8 37 52 77)(9 38 53 78)(10 39 54 79)(11 40 55 80)(12 41 56 81)(13 96 65 33)(14 85 66 34)(15 86 67 35)(16 87 68 36)(17 88 69 25)(18 89 70 26)(19 90 71 27)(20 91 72 28)(21 92 61 29)(22 93 62 30)(23 94 63 31)(24 95 64 32)
(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 7 12)(2 11 8 5)(3 4 9 10)(13 22 19 16)(14 15 20 21)(17 18 23 24)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 40 43 46)(38 45 44 39)(41 48 47 42)(49 58 55 52)(50 51 56 57)(53 54 59 60)(61 66 67 72)(62 71 68 65)(63 64 69 70)(73 84 79 78)(74 77 80 83)(75 82 81 76)(85 92 91 86)(87 90 93 96)(88 95 94 89)```

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4N 4O ··· 4AB 6A 6B 6C 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 6 6 2 2 ··· 2 6 ··· 6 2 2 2 2 2 2 2 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 C4×S3 2- 1+4 Q8.15D6 Q8○D12 kernel C42.125D6 C4×Dic6 C42⋊2S3 Dic6⋊C4 C4⋊C4⋊7S3 Q8×Dic3 Q8×C12 C2×S3×Q8 S3×Q8 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C6 C2 C2 # reps 1 3 3 3 3 1 1 1 16 1 3 3 1 8 2 2 2

Matrix representation of C42.125D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 12 0 0 0 0 1 0 12 0 0 2 0 12 0 0 0 0 2 0 12
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 11 4 0 0 0 0 9 2 0 0 0 0 0 0 11 4 0 0 0 0 9 2
,
 1 1 0 0 0 0 12 0 0 0 0 0 0 0 0 5 0 8 0 0 8 5 5 8 0 0 0 0 0 8 0 0 0 0 5 8
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 5 0 8 0 0 0 5 8 8 5 0 0 0 0 8 0 0 0 0 0 8 5

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,2,0,0,0,0,1,0,2,0,0,12,0,12,0,0,0,0,12,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,5,5,0,0,0,0,0,5,0,5,0,0,8,8,8,8],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,5,5,0,0,0,0,0,8,0,0,0,0,8,8,8,8,0,0,0,5,0,5] >;`

C42.125D6 in GAP, Magma, Sage, TeX

`C_4^2._{125}D_6`
`% in TeX`

`G:=Group("C4^2.125D6");`
`// GroupNames label`

`G:=SmallGroup(192,1131);`
`// by ID`

`G=gap.SmallGroup(192,1131);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,1123,80,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;`
`// generators/relations`

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