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## G = C42.270D6order 192 = 26·3

### 29th non-split extension by C42 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42.270D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C12.55D4 — C42.270D6
 Lower central C3 — C2×C6 — C42.270D6
 Upper central C1 — C2×C4 — C2×C42

Generators and relations for C42.270D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b, ab=ba, ac=ca, dad-1=a-1b2, bc=cb, bd=db, dcd-1=a2c-1 >

Subgroups: 184 in 110 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C2×C8, C22×C4, C3⋊C8, C2×C12, C2×C12, C22×C6, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×C12, C22×C12, C42.6C4, C42.S3, C12⋊C8, C12.55D4, C2×C4×C12, C42.270D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, M4(2), C22×C4, C4○D4, C2×Dic3, C22×S3, C42⋊C2, C2×M4(2), C4.Dic3, C4○D12, C22×Dic3, C42.6C4, C2×C4.Dic3, C23.26D6, C42.270D6

Smallest permutation representation of C42.270D6
On 96 points
Generators in S96
```(1 51 76 12)(2 9 77 56)(3 53 78 14)(4 11 79 50)(5 55 80 16)(6 13 73 52)(7 49 74 10)(8 15 75 54)(17 57 31 41)(18 46 32 62)(19 59 25 43)(20 48 26 64)(21 61 27 45)(22 42 28 58)(23 63 29 47)(24 44 30 60)(33 87 70 95)(34 92 71 84)(35 81 72 89)(36 94 65 86)(37 83 66 91)(38 96 67 88)(39 85 68 93)(40 90 69 82)
(1 78 5 74)(2 79 6 75)(3 80 7 76)(4 73 8 77)(9 50 13 54)(10 51 14 55)(11 52 15 56)(12 53 16 49)(17 25 21 29)(18 26 22 30)(19 27 23 31)(20 28 24 32)(33 72 37 68)(34 65 38 69)(35 66 39 70)(36 67 40 71)(41 59 45 63)(42 60 46 64)(43 61 47 57)(44 62 48 58)(81 91 85 95)(82 92 86 96)(83 93 87 89)(84 94 88 90)
(1 95 62)(2 47 96 77 63 88)(3 89 64)(4 41 90 79 57 82)(5 91 58)(6 43 92 73 59 84)(7 93 60)(8 45 94 75 61 86)(9 23 67 56 29 38)(10 68 30)(11 17 69 50 31 40)(12 70 32)(13 19 71 52 25 34)(14 72 26)(15 21 65 54 27 36)(16 66 28)(18 51 33)(20 53 35)(22 55 37)(24 49 39)(42 80 83)(44 74 85)(46 76 87)(48 78 81)
(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)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E ··· 4N 6A ··· 6G 8A ··· 8H 12A ··· 12X order 1 2 2 2 2 2 3 4 4 4 4 4 ··· 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 1 1 2 2 2 1 1 1 1 2 ··· 2 2 ··· 2 12 ··· 12 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 S3 Dic3 D6 Dic3 D6 M4(2) C4○D4 M4(2) C4.Dic3 C4○D12 C4.Dic3 kernel C42.270D6 C42.S3 C12⋊C8 C12.55D4 C2×C4×C12 C4×C12 C22×C12 C2×C42 C42 C42 C22×C4 C22×C4 C12 C12 C2×C6 C4 C4 C22 # reps 1 2 2 2 1 4 4 1 2 2 2 1 4 4 4 8 8 8

Matrix representation of C42.270D6 in GL4(𝔽73) generated by

 46 34 0 0 0 27 0 0 0 0 46 0 0 0 0 46
,
 72 0 0 0 0 72 0 0 0 0 46 0 0 0 0 46
,
 8 52 0 0 0 9 0 0 0 0 1 0 0 0 0 72
,
 65 8 0 0 56 8 0 0 0 0 0 1 0 0 27 0
`G:=sub<GL(4,GF(73))| [46,0,0,0,34,27,0,0,0,0,46,0,0,0,0,46],[72,0,0,0,0,72,0,0,0,0,46,0,0,0,0,46],[8,0,0,0,52,9,0,0,0,0,1,0,0,0,0,72],[65,56,0,0,8,8,0,0,0,0,0,27,0,0,1,0] >;`

C42.270D6 in GAP, Magma, Sage, TeX

`C_4^2._{270}D_6`
`% in TeX`

`G:=Group("C4^2.270D6");`
`// GroupNames label`

`G:=SmallGroup(192,485);`
`// by ID`

`G=gap.SmallGroup(192,485);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,253,758,100,136,6278]);`
`// Polycyclic`

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

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