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## G = C42.108D6order 192 = 26·3

### 108th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42.108D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×D4⋊2S3 — C42.108D6
 Lower central C3 — C6 — C42.108D6
 Upper central C1 — C22 — C4×D4

Generators and relations for C42.108D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=a2c-1 >

Subgroups: 616 in 294 conjugacy classes, 151 normal (43 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×14], C22, C22 [×4], C22 [×8], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×25], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, Dic3 [×6], Dic3 [×4], C12 [×2], C12 [×4], D6 [×2], D6 [×2], C2×C6, C2×C6 [×4], C2×C6 [×4], C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], Dic6 [×4], C4×S3 [×4], C4×S3 [×2], C2×Dic3 [×3], C2×Dic3 [×12], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×3], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C22×S3, C22×C6 [×2], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4, C4×D4 [×5], C4×Q8 [×2], C2×C4○D4, C4×Dic3, C4×Dic3 [×4], Dic3⋊C4 [×2], Dic3⋊C4 [×6], C4⋊Dic3, D6⋊C4 [×2], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4 [×2], D42S3 [×8], C22×Dic3 [×4], C2×C3⋊D4 [×2], C22×C12 [×2], C6×D4, C23.33C23, C4×Dic6, C422S3, C23.16D6 [×2], Dic34D4 [×2], Dic6⋊C4, S3×C4⋊C4, C2×Dic3⋊C4 [×2], C4×C3⋊D4 [×2], D4×Dic3, D4×C12, C2×D42S3, C42.108D6
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, 2+ 1+4, 2- 1+4, S3×C2×C4 [×6], S3×C23, C23.33C23, S3×C22×C4, D46D6, Q8○D12, C42.108D6

Smallest permutation representation of C42.108D6
On 96 points
Generators in S96
```(1 76 37 85)(2 86 38 77)(3 78 39 87)(4 88 40 73)(5 74 41 89)(6 90 42 75)(7 65 50 27)(8 28 51 66)(9 61 52 29)(10 30 53 62)(11 63 54 25)(12 26 49 64)(13 84 45 93)(14 94 46 79)(15 80 47 95)(16 96 48 81)(17 82 43 91)(18 92 44 83)(19 70 33 55)(20 56 34 71)(21 72 35 57)(22 58 36 67)(23 68 31 59)(24 60 32 69)
(1 10 31 14)(2 11 32 15)(3 12 33 16)(4 7 34 17)(5 8 35 18)(6 9 36 13)(19 48 39 49)(20 43 40 50)(21 44 41 51)(22 45 42 52)(23 46 37 53)(24 47 38 54)(25 60 95 77)(26 55 96 78)(27 56 91 73)(28 57 92 74)(29 58 93 75)(30 59 94 76)(61 67 84 90)(62 68 79 85)(63 69 80 86)(64 70 81 87)(65 71 82 88)(66 72 83 89)
(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 37 42)(2 41 38 5)(3 4 39 40)(7 12 50 49)(8 54 51 11)(9 10 52 53)(13 14 45 46)(15 18 47 44)(16 43 48 17)(19 20 33 34)(21 24 35 32)(22 31 36 23)(25 28 63 66)(26 65 64 27)(29 30 61 62)(55 56 70 71)(57 60 72 69)(58 68 67 59)(73 87 88 78)(74 77 89 86)(75 85 90 76)(79 93 94 84)(80 83 95 92)(81 91 96 82)```

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 D6 D6 C4×S3 2+ 1+4 2- 1+4 D4⋊6D6 Q8○D12 kernel C42.108D6 C4×Dic6 C42⋊2S3 C23.16D6 Dic3⋊4D4 Dic6⋊C4 S3×C4⋊C4 C2×Dic3⋊C4 C4×C3⋊D4 D4×Dic3 D4×C12 C2×D4⋊2S3 D4⋊2S3 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 D4 C6 C6 C2 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 16 1 1 2 1 2 1 8 1 1 2 2

Matrix representation of C42.108D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 3 0 0 0 0 3 9 0 0 0 0 0 0 9 10 0 0 0 0 10 4
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 4 0 0 3 0 0 0 4 10 0 0 0 0 10 9 0 0 0 3 0 0 9
,
 1 1 0 0 0 0 12 0 0 0 0 0 0 0 0 9 3 0 0 0 4 0 0 3 0 0 10 0 0 4 0 0 0 10 9 0
,
 12 0 0 0 0 0 1 1 0 0 0 0 0 0 0 4 10 0 0 0 9 0 0 10 0 0 8 0 0 9 0 0 0 8 4 0

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,3,9,0,0,0,0,0,0,9,10,0,0,0,0,10,4],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,4,0,0,3,0,0,0,4,10,0,0,0,0,10,9,0,0,0,3,0,0,9],[1,12,0,0,0,0,1,0,0,0,0,0,0,0,0,4,10,0,0,0,9,0,0,10,0,0,3,0,0,9,0,0,0,3,4,0],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,0,9,8,0,0,0,4,0,0,8,0,0,10,0,0,4,0,0,0,10,9,0] >;`

C42.108D6 in GAP, Magma, Sage, TeX

`C_4^2._{108}D_6`
`% in TeX`

`G:=Group("C4^2.108D6");`
`// GroupNames label`

`G:=SmallGroup(192,1105);`
`// by ID`

`G=gap.SmallGroup(192,1105);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,570,80,6278]);`
`// Polycyclic`

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

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