Copied to
clipboard

## G = C42.65D6order 192 = 26·3

### 65th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.65D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12⋊2Q8 — C42.65D6
 Lower central C3 — C6 — C2×C12 — C42.65D6
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.65D6
G = < a,b,c,d | a4=b4=c6=1, d2=b, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 336 in 124 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6, C6 [×2], C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×6], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], Q16 [×4], C2×D4, C2×Q8, C2×Q8 [×2], C3⋊C8 [×4], Dic6 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, C8⋊C4, C4.4D4, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], D4.S3 [×4], C3⋊Q16 [×4], C4×C12, C3×C22⋊C4 [×2], C2×Dic6 [×2], C6×D4, C6×Q8, C8.2D4, C42.S3, C122Q8, C2×D4.S3 [×2], C2×C3⋊Q16 [×2], C3×C4.4D4, C42.65D6
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, Q8.14D6 [×2], C42.65D6

Character table of C42.65D6

 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 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 ρ10 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 -2 2 -2 0 0 0 -2 0 2 -2 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ11 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 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 S3 ρ13 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 -2 2 -2 0 0 -2 0 2 0 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ14 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 ρ15 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 -2 2 -2 0 0 2 0 -2 0 2 0 0 -2 0 0 0 0 orthogonal lifted from D4 ρ16 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 ρ17 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 -2 2 -2 0 0 0 2 0 -2 -2 0 0 2 0 0 0 0 orthogonal lifted from D4 ρ18 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 ρ19 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 ρ20 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 ρ21 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 ρ22 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 ρ23 4 -4 -4 4 0 -2 4 -4 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 -2 0 0 2 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 0 2 0 0 -2 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 0 0 0 2√3 -2√3 0 0 0 0 0 symplectic lifted from Q8.14D6, 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 0 -2√3 2√3 0 0 0 0 0 symplectic lifted from Q8.14D6, Schur index 2 ρ29 4 -4 4 -4 0 -2 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 2√3 -2√3 0 0 symplectic lifted from Q8.14D6, 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 0 0 0 0 -2√3 2√3 0 0 symplectic lifted from Q8.14D6, Schur index 2

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

Matrix representation of C42.65D6 in GL8(𝔽73)

 15 28 69 0 0 0 0 0 45 60 0 69 0 0 0 0 43 14 58 45 0 0 0 0 59 29 28 13 0 0 0 0 0 0 0 0 0 0 66 59 0 0 0 0 0 0 14 7 0 0 0 0 7 14 0 0 0 0 0 0 59 66 0 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0
,
 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 43 29 1 1 0 0 0 0 44 14 72 0 0 0 0 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 72 0
,
 42 45 0 0 0 0 0 0 3 31 0 0 0 0 0 0 44 19 42 45 0 0 0 0 48 29 3 31 0 0 0 0 0 0 0 0 69 28 4 45 0 0 0 0 32 4 41 69 0 0 0 0 69 28 69 28 0 0 0 0 32 4 32 4

`G:=sub<GL(8,GF(73))| [15,45,43,59,0,0,0,0,28,60,14,29,0,0,0,0,69,0,58,28,0,0,0,0,0,69,45,13,0,0,0,0,0,0,0,0,0,0,7,59,0,0,0,0,0,0,14,66,0,0,0,0,66,14,0,0,0,0,0,0,59,7,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[72,1,43,44,0,0,0,0,72,0,29,14,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,72,0,0,0,0,0,0,1,0],[42,3,44,48,0,0,0,0,45,31,19,29,0,0,0,0,0,0,42,3,0,0,0,0,0,0,45,31,0,0,0,0,0,0,0,0,69,32,69,32,0,0,0,0,28,4,28,4,0,0,0,0,4,41,69,32,0,0,0,0,45,69,28,4] >;`

C42.65D6 in GAP, Magma, Sage, TeX

`C_4^2._{65}D_6`
`% in TeX`

`G:=Group("C4^2.65D6");`
`// GroupNames label`

`G:=SmallGroup(192,619);`
`// by ID`

`G=gap.SmallGroup(192,619);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,555,1123,297,136,6278]);`
`// Polycyclic`

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

Export

׿
×
𝔽