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## G = C42.82D6order 192 = 26·3

### 82nd non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.82D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C42⋊7S3 — C42.82D6
 Lower central C3 — C6 — C2×C12 — C42.82D6
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C42.82D6
G = < a,b,c,d | a4=b4=1, c6=a2b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 304 in 100 conjugacy classes, 39 normal (25 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], S3, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×4], C23, Dic3, C12 [×2], C12 [×4], D6 [×3], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×D4, C2×Q8 [×2], C3⋊C8 [×2], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C22×S3, C8⋊C4, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C4⋊Q8, C2×C3⋊C8 [×2], D6⋊C4 [×2], C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C2×D12, C6×Q8, C42.28C22, C42.S3, C6.D8 [×2], C6.SD16 [×2], C427S3, C3×C4⋊Q8, C42.82D6
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, Q83S3 [×2], C2×C3⋊D4, C42.28C22, D126C22, Q8.11D6, C12.23D4, C42.82D6

Character table of C42.82D6

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J size 1 1 1 1 24 2 2 2 4 4 8 8 24 2 2 2 12 12 12 12 4 4 4 4 4 4 8 8 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 0 2 -2 -2 2 -2 0 0 0 2 2 2 0 0 0 0 -2 -2 -2 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 0 -1 2 2 -2 -2 2 -2 0 -1 -1 -1 0 0 0 0 1 -1 -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 2 -2 -2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -1 2 2 2 2 -2 -2 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ13 2 2 2 2 0 -1 2 2 2 2 2 2 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ14 2 2 2 2 0 -1 2 2 -2 -2 -2 2 0 -1 -1 -1 0 0 0 0 1 -1 -1 1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ15 2 2 2 2 0 -1 -2 -2 2 -2 0 0 0 -1 -1 -1 0 0 0 0 1 1 1 -1 -1 1 -√-3 -√-3 √-3 √-3 complex lifted from C3⋊D4 ρ16 2 2 2 2 0 -1 -2 -2 -2 2 0 0 0 -1 -1 -1 0 0 0 0 -1 1 1 1 1 -1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ17 2 2 2 2 0 -1 -2 -2 -2 2 0 0 0 -1 -1 -1 0 0 0 0 -1 1 1 1 1 -1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 2 2 0 -1 -2 -2 2 -2 0 0 0 -1 -1 -1 0 0 0 0 1 1 1 -1 -1 1 √-3 √-3 -√-3 -√-3 complex lifted from C3⋊D4 ρ19 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 -2 -2 2 2i 0 -2i 0 0 -2 2 0 0 0 0 0 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 2i 0 -2i 0 2 -2 0 0 0 0 0 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 -2i 0 2i 0 2 -2 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 -2 -2 2 -2i 0 2i 0 0 -2 2 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 -4 4 0 -2 4 -4 0 0 0 0 0 2 2 -2 0 0 0 0 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 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 -4 4 0 0 0 0 0 2 2 -2 0 0 0 0 0 -2 2 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ26 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 ρ27 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 2√-3 0 0 0 0 -2√-3 0 0 0 0 complex lifted from D12⋊6C22 ρ28 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 complex lifted from Q8.11D6 ρ29 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 complex lifted from Q8.11D6 ρ30 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 -2√-3 0 0 0 0 2√-3 0 0 0 0 complex lifted from D12⋊6C22

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

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

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

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

Matrix representation of C42.82D6 in GL6(𝔽73)

 54 6 0 0 0 0 37 19 0 0 0 0 0 0 0 0 30 60 0 0 0 0 13 43 0 0 43 13 0 0 0 0 60 30 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
,
 27 0 0 0 0 0 25 46 0 0 0 0 0 0 42 31 31 42 0 0 42 11 31 62 0 0 31 42 31 42 0 0 31 62 31 62
,
 27 0 0 0 0 0 0 27 0 0 0 0 0 0 31 42 31 42 0 0 11 42 11 42 0 0 42 31 31 42 0 0 62 31 11 42

`G:=sub<GL(6,GF(73))| [54,37,0,0,0,0,6,19,0,0,0,0,0,0,0,0,43,60,0,0,0,0,13,30,0,0,30,13,0,0,0,0,60,43,0,0],[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],[27,25,0,0,0,0,0,46,0,0,0,0,0,0,42,42,31,31,0,0,31,11,42,62,0,0,31,31,31,31,0,0,42,62,42,62],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,31,11,42,62,0,0,42,42,31,31,0,0,31,11,31,11,0,0,42,42,42,42] >;`

C42.82D6 in GAP, Magma, Sage, TeX

`C_4^2._{82}D_6`
`% in TeX`

`G:=Group("C4^2.82D6");`
`// GroupNames label`

`G:=SmallGroup(192,648);`
`// by ID`

`G=gap.SmallGroup(192,648);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,100,1123,297,136,6278]);`
`// Polycyclic`

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

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