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## G = C42.80D6order 192 = 26·3

### 80th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.80D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C42⋊7S3 — C42.80D6
 Lower central C3 — C6 — C2×C12 — C42.80D6
 Upper central C1 — C22 — C42 — C4⋊Q8

Generators and relations for C42.80D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 368 in 124 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], S3, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, Dic3, C12 [×2], C12 [×4], D6 [×3], C2×C6, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], Q16 [×4], C2×D4, C2×Q8 [×2], C2×Q8, C3⋊C8 [×4], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C2×C12 [×2], C3×Q8 [×4], C22×S3, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C2×C3⋊C8 [×2], D6⋊C4 [×2], Q82S3 [×4], C3⋊Q16 [×4], C4×C12, C3×C4⋊C4 [×2], C2×Dic6, C2×D12, C6×Q8 [×2], C8.2D4, C42.S3, C427S3, C2×Q82S3 [×2], C2×C3⋊Q16 [×2], C3×C4⋊Q8, C42.80D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C8.C22 [×2], S3×D4 [×2], C2×C3⋊D4, C8.2D4, C123D4, Q8.11D6 [×2], C42.80D6

Character table of C42.80D6

 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 2 2 -2 0 0 0 0 0 -2 -2 2 2 0 -2 0 0 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 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 ρ12 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 ρ13 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 -2 -2 2 0 2 0 -2 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 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 2 2 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ16 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 -2 -2 2 0 -2 0 2 0 2 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ17 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 ρ18 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 -2 -2 2 -2 0 2 0 0 -2 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 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 ρ20 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 ρ21 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 ρ22 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 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 S3×D4 ρ24 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 S3×D4 ρ25 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 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 Q8.11D6 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 Q8.11D6

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

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

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

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

Matrix representation of C42.80D6 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 30 60 60 47 0 0 13 43 26 13 0 0 43 13 43 13 0 0 60 30 60 30
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 71 0 0 0 0 72 0 71 0 0 1 0 1 0 0 0 0 1 0 1
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 69 4 47 55 0 0 69 65 18 65 0 0 64 60 4 69 0 0 13 4 4 8
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 22 22 18 26 0 0 0 51 8 55 0 0 64 60 4 69 0 0 69 9 65 69

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,30,13,43,60,0,0,60,43,13,30,0,0,60,26,43,60,0,0,47,13,13,30],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,1,0,0,0,0,72,0,1,0,0,71,0,1,0,0,0,0,71,0,1],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,69,69,64,13,0,0,4,65,60,4,0,0,47,18,4,4,0,0,55,65,69,8],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,22,0,64,69,0,0,22,51,60,9,0,0,18,8,4,65,0,0,26,55,69,69] >;`

C42.80D6 in GAP, Magma, Sage, TeX

`C_4^2._{80}D_6`
`% in TeX`

`G:=Group("C4^2.80D6");`
`// GroupNames label`

`G:=SmallGroup(192,645);`
`// by ID`

`G=gap.SmallGroup(192,645);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,184,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^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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