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## G = C42.282D6order 192 = 26·3

### 5th central extension by C42 of D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C42.282D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — S3×C42 — C42.282D6
 Lower central C3 — C6 — C42.282D6
 Upper central C1 — C42 — C4×C8

Generators and relations for C42.282D6
G = < a,b,c,d | a4=b4=1, c6=b-1, d2=a2b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2b2c5 >

Subgroups: 248 in 118 conjugacy classes, 63 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, C42.12C4, C4×C3⋊C8, Dic3⋊C8, D6⋊C8, C4×C24, S3×C42, C42.282D6
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, M4(2), C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C22×C8, C2×M4(2), S3×C8, C8⋊S3, S3×C2×C4, C4○D12, C42.12C4, C422S3, S3×C2×C8, C2×C8⋊S3, C42.282D6

Smallest permutation representation of C42.282D6
On 96 points
Generators in S96
```(1 38 54 88)(2 39 55 89)(3 40 56 90)(4 41 57 91)(5 42 58 92)(6 43 59 93)(7 44 60 94)(8 45 61 95)(9 46 62 96)(10 47 63 73)(11 48 64 74)(12 25 65 75)(13 26 66 76)(14 27 67 77)(15 28 68 78)(16 29 69 79)(17 30 70 80)(18 31 71 81)(19 32 72 82)(20 33 49 83)(21 34 50 84)(22 35 51 85)(23 36 52 86)(24 37 53 87)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 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 18 72 65 13 6 60 53)(2 64 49 5 14 52 61 17)(3 4 50 51 15 16 62 63)(7 24 54 71 19 12 66 59)(8 70 55 11 20 58 67 23)(9 10 56 57 21 22 68 69)(25 76 93 44 37 88 81 32)(26 43 94 87 38 31 82 75)(27 86 95 30 39 74 83 42)(28 29 96 73 40 41 84 85)(33 92 77 36 45 80 89 48)(34 35 78 79 46 47 90 91)```

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A ··· 4L 4M ··· 4R 6A 6B 6C 8A ··· 8H 8I ··· 8P 12A ··· 12L 24A ··· 24P order 1 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 6 6 8 ··· 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 6 6 2 1 ··· 1 6 ··· 6 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C8 S3 D6 D6 M4(2) C4○D4 C4×S3 S3×C8 C8⋊S3 C4○D12 kernel C42.282D6 C4×C3⋊C8 Dic3⋊C8 D6⋊C8 C4×C24 S3×C42 C4×Dic3 S3×C2×C4 C4×S3 C4×C8 C42 C2×C8 C12 C12 C2×C4 C4 C4 C4 # reps 1 1 2 2 1 1 4 4 16 1 1 2 4 4 4 8 8 8

Matrix representation of C42.282D6 in GL3(𝔽73) generated by

 1 0 0 0 46 0 0 0 46
,
 46 0 0 0 1 0 0 0 1
,
 51 0 0 0 43 30 0 43 13
,
 22 0 0 0 30 43 0 13 43
`G:=sub<GL(3,GF(73))| [1,0,0,0,46,0,0,0,46],[46,0,0,0,1,0,0,0,1],[51,0,0,0,43,43,0,30,13],[22,0,0,0,30,13,0,43,43] >;`

C42.282D6 in GAP, Magma, Sage, TeX

`C_4^2._{282}D_6`
`% in TeX`

`G:=Group("C4^2.282D6");`
`// GroupNames label`

`G:=SmallGroup(192,244);`
`// by ID`

`G=gap.SmallGroup(192,244);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,422,58,136,6278]);`
`// Polycyclic`

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

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