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## G = C42.62D6order 192 = 26·3

### 62nd non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.62D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C42.S3 — C42.62D6
 Lower central C3 — C6 — C2×C12 — C42.62D6
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.62D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2bc-1 >

Subgroups: 272 in 100 conjugacy classes, 39 normal (27 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6 [×3], C6, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×2], C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×2], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C22⋊C4 [×2], C2×Dic6, C6×D4, C6×Q8, C42.28C22, C42.S3, D4⋊Dic3 [×2], Q82Dic3 [×2], C122Q8, C3×C4.4D4, C42.62D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C4.4D4, C8⋊C22, C8.C22, D42S3 [×2], C2×C3⋊D4, C42.28C22, C23.12D6, D4⋊D6, Q8.14D6, C42.62D6

Character table of C42.62D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 8 2 2 2 4 4 8 24 24 2 2 2 8 8 12 12 12 12 4 4 4 4 4 4 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 -1 1 1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ4 1 1 1 1 -1 1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ7 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ9 2 2 2 2 -2 -1 2 2 -2 -2 2 0 0 -1 -1 -1 1 1 0 0 0 0 1 1 1 1 -1 -1 -1 -1 orthogonal lifted from D6 ρ10 2 2 2 2 2 -1 2 2 -2 -2 -2 0 0 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 0 2 -2 -2 2 -2 0 0 0 2 2 2 0 0 0 0 0 0 2 -2 -2 2 -2 -2 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 -2 -1 2 2 2 2 -2 0 0 -1 -1 -1 1 1 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 orthogonal lifted from D6 ρ13 2 2 2 2 2 -1 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 2 0 2 -2 -2 -2 2 0 0 0 2 2 2 0 0 0 0 0 0 -2 2 2 -2 -2 -2 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 0 -1 -2 -2 -2 2 0 0 0 -1 -1 -1 √-3 -√-3 0 0 0 0 1 -1 -1 1 1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ16 2 2 2 2 0 -1 -2 -2 2 -2 0 0 0 -1 -1 -1 -√-3 √-3 0 0 0 0 -1 1 1 -1 1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ17 2 2 2 2 0 -1 -2 -2 2 -2 0 0 0 -1 -1 -1 √-3 -√-3 0 0 0 0 -1 1 1 -1 1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 2 2 0 -1 -2 -2 -2 2 0 0 0 -1 -1 -1 -√-3 √-3 0 0 0 0 1 -1 -1 1 1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ19 2 -2 2 -2 0 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 0 2i -2i 0 0 0 0 0 2 -2 0 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 0 2 -2 2 0 0 0 0 0 2 -2 -2 0 0 -2i 0 0 2i 0 0 0 0 -2 2 0 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 0 2 -2 2 0 0 0 0 0 2 -2 -2 0 0 2i 0 0 -2i 0 0 0 0 -2 2 0 0 complex lifted from C4○D4 ρ22 2 -2 2 -2 0 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 0 -2i 2i 0 0 0 0 0 2 -2 0 0 complex lifted from C4○D4 ρ23 4 4 -4 -4 0 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 2√3 -2√3 0 0 0 0 0 orthogonal lifted from D4⋊D6 ρ24 4 4 -4 -4 0 4 0 0 0 0 0 0 0 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ25 4 4 -4 -4 0 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 -2√3 2√3 0 0 0 0 0 orthogonal lifted from D4⋊D6 ρ26 4 -4 4 -4 0 -2 -4 4 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ27 4 -4 -4 4 0 4 0 0 0 0 0 0 0 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from C8.C22, Schur index 2 ρ28 4 -4 -4 4 0 -2 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 2√3 0 0 -2√3 0 0 0 0 symplectic lifted from Q8.14D6, Schur index 2 ρ29 4 -4 4 -4 0 -2 4 -4 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 symplectic lifted from D4⋊2S3, Schur index 2 ρ30 4 -4 -4 4 0 -2 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 -2√3 0 0 2√3 0 0 0 0 symplectic lifted from Q8.14D6, Schur index 2

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

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

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

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

Matrix representation of C42.62D6 in GL6(𝔽73)

 21 3 0 0 0 0 23 52 0 0 0 0 0 0 0 0 66 59 0 0 0 0 14 7 0 0 7 14 0 0 0 0 59 66 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 72 0 0
,
 1 0 0 0 0 0 59 72 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 72 0
,
 17 65 0 0 0 0 18 56 0 0 0 0 0 0 69 3 69 3 0 0 7 4 7 4 0 0 69 3 4 70 0 0 7 4 66 69

`G:=sub<GL(6,GF(73))| [21,23,0,0,0,0,3,52,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,66,14,0,0,0,0,59,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[1,59,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[17,18,0,0,0,0,65,56,0,0,0,0,0,0,69,7,69,7,0,0,3,4,3,4,0,0,69,7,4,66,0,0,3,4,70,69] >;`

C42.62D6 in GAP, Magma, Sage, TeX

`C_4^2._{62}D_6`
`% in TeX`

`G:=Group("C4^2.62D6");`
`// GroupNames label`

`G:=SmallGroup(192,614);`
`// by ID`

`G=gap.SmallGroup(192,614);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,590,471,438,102,6278]);`
`// 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=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*b*c^-1>;`
`// generators/relations`

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