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## G = C42.276D6order 192 = 26·3

### 35th non-split extension by C42 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42.276D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — C2×D12 — C4×D12 — C42.276D6
 Lower central C3 — C2×C6 — C42.276D6
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C42.276D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, bc=cb, bd=db, dcd-1=b2c-1 >

Subgroups: 824 in 310 conjugacy classes, 119 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×8], C4 [×6], C22, C22 [×2], C22 [×14], S3 [×4], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×16], D4 [×20], Q8 [×4], C23, C23 [×4], Dic3 [×4], C12 [×8], C12 [×2], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic6 [×4], C4×S3 [×8], D12 [×12], C2×Dic3 [×4], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×8], C2×C12 [×4], C22×S3 [×4], C22×C6, C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4 [×2], C4⋊Dic3 [×4], D6⋊C4 [×8], C4×C12 [×2], C4×C12 [×2], C2×Dic6 [×2], S3×C2×C4 [×4], C2×D12 [×6], C4○D12 [×8], C2×C3⋊D4 [×4], C22×C12, C22×C12 [×2], C22.26C24, C122Q8, C4×D12 [×4], C4⋊D12, C427S3 [×2], C127D4 [×4], C2×C4×C12, C2×C4○D12 [×2], C42.276D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C2×D12 [×6], C4○D12 [×4], S3×C23, C22.26C24, C22×D12, C2×C4○D12 [×2], C42.276D6

Smallest permutation representation of C42.276D6
On 96 points
Generators in S96
```(1 67 19 59)(2 68 20 60)(3 69 21 55)(4 70 22 56)(5 71 23 57)(6 72 24 58)(7 75 27 62)(8 76 28 63)(9 77 29 64)(10 78 30 65)(11 73 25 66)(12 74 26 61)(13 51 86 38)(14 52 87 39)(15 53 88 40)(16 54 89 41)(17 49 90 42)(18 50 85 37)(31 93 45 79)(32 94 46 80)(33 95 47 81)(34 96 48 82)(35 91 43 83)(36 92 44 84)
(1 40 9 35)(2 41 10 36)(3 42 11 31)(4 37 12 32)(5 38 7 33)(6 39 8 34)(13 75 95 71)(14 76 96 72)(15 77 91 67)(16 78 92 68)(17 73 93 69)(18 74 94 70)(19 53 29 43)(20 54 30 44)(21 49 25 45)(22 50 26 46)(23 51 27 47)(24 52 28 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)
(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 58 9 63)(2 62 10 57)(3 56 11 61)(4 66 12 55)(5 60 7 65)(6 64 8 59)(13 54 95 44)(14 43 96 53)(15 52 91 48)(16 47 92 51)(17 50 93 46)(18 45 94 49)(19 72 29 76)(20 75 30 71)(21 70 25 74)(22 73 26 69)(23 68 27 78)(24 77 28 67)(31 80 42 85)(32 90 37 79)(33 84 38 89)(34 88 39 83)(35 82 40 87)(36 86 41 81)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 6A ··· 6G 12A ··· 12X order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 12 12 12 12 2 1 1 1 1 2 ··· 2 12 12 12 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 D12 C4○D12 kernel C42.276D6 C12⋊2Q8 C4×D12 C4⋊D12 C42⋊7S3 C12⋊7D4 C2×C4×C12 C2×C4○D12 C2×C42 C2×C12 C42 C22×C4 C12 C2×C4 C4 # reps 1 1 4 1 2 4 1 2 1 4 4 3 8 8 16

Matrix representation of C42.276D6 in GL4(𝔽13) generated by

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

C42.276D6 in GAP, Magma, Sage, TeX

`C_4^2._{276}D_6`
`% in TeX`

`G:=Group("C4^2.276D6");`
`// GroupNames label`

`G:=SmallGroup(192,1036);`
`// by ID`

`G=gap.SmallGroup(192,1036);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,100,675,6278]);`
`// Polycyclic`

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

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