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## G = C3⋊C8.6D4order 192 = 26·3

### 6th non-split extension by C3⋊C8 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C3⋊C8.6D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×Dic6 — C12.48D4 — C3⋊C8.6D4
 Lower central C3 — C6 — C2×C12 — C3⋊C8.6D4
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for C3⋊C8.6D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b3, dbd=b5, dcd=b4c-1 >

Subgroups: 272 in 110 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×5], C22, C22 [×3], C6 [×3], C6, C8 [×3], C2×C4 [×2], C2×C4 [×6], Q8 [×4], C23, Dic3 [×2], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], Q16 [×2], C22×C4, C2×Q8, C2×Q8, C3⋊C8 [×2], C3⋊C8, Dic6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C3×Q8 [×2], C22×C6, Q8⋊C4 [×2], C4.Q8, C22⋊Q8, C22⋊Q8, C2×M4(2), C2×Q16, C2×C3⋊C8 [×2], C4.Dic3 [×2], Dic3⋊C4, C4⋊Dic3, C3⋊Q16 [×2], C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×Q8, C8.D4, C12.Q8, C6.SD16, Q82Dic3, C2×C4.Dic3, C12.48D4, C2×C3⋊Q16, C3×C22⋊Q8, C3⋊C8.6D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C8.C22 [×2], S3×D4, D42S3, C2×C3⋊D4, C8.D4, C23.14D6, Q8.11D6, Q8.14D6, C3⋊C8.6D4

Character table of C3⋊C8.6D4

 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 4 2 2 2 4 8 8 24 24 2 2 2 4 4 12 12 12 12 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 0 0 0 0 0 2 -2 -2 0 0 0 -2 2 0 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 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 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 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 D6 ρ14 2 2 2 2 -2 2 -2 -2 2 0 0 0 0 2 2 2 -2 -2 0 0 0 0 -2 2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 0 2 -2 2 0 0 0 0 0 2 -2 -2 0 0 0 2 -2 0 2 0 -2 0 0 0 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 2 2 -2 -2 -2 0 0 0 0 2 2 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 2 2 2 -1 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 2 2 -2 -1 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 0 0 0 0 1 -1 1 -1 -√-3 -√-3 √-3 √-3 complex lifted from C3⋊D4 ρ19 2 2 2 2 2 -1 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ20 2 2 2 2 -2 -1 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 0 0 0 0 1 -1 1 -1 √-3 √-3 -√-3 -√-3 complex lifted from C3⋊D4 ρ21 2 2 -2 -2 0 2 2 -2 0 0 0 0 0 2 -2 -2 0 0 2i 0 0 -2i -2 0 2 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 0 0 -2i 0 0 2i -2 0 2 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 0 -2 0 2 0 0 0 0 0 orthogonal lifted from S3×D4 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 symplectic lifted from C8.C22, Schur index 2 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 -2 4 -4 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 2 0 -2 0 0 0 0 0 symplectic lifted from D4⋊2S3, 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 0 -2√3 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 0 2√3 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 2√-3 -2√-3 0 0 0 0 0 0 0 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 -2√-3 2√-3 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from Q8.11D6

Smallest permutation representation of C3⋊C8.6D4
On 96 points
Generators in S96
```(1 83 94)(2 95 84)(3 85 96)(4 89 86)(5 87 90)(6 91 88)(7 81 92)(8 93 82)(9 45 55)(10 56 46)(11 47 49)(12 50 48)(13 41 51)(14 52 42)(15 43 53)(16 54 44)(17 31 33)(18 34 32)(19 25 35)(20 36 26)(21 27 37)(22 38 28)(23 29 39)(24 40 30)(57 73 68)(58 69 74)(59 75 70)(60 71 76)(61 77 72)(62 65 78)(63 79 66)(64 67 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 57 12 24)(2 60 13 19)(3 63 14 22)(4 58 15 17)(5 61 16 20)(6 64 9 23)(7 59 10 18)(8 62 11 21)(25 84 71 51)(26 87 72 54)(27 82 65 49)(28 85 66 52)(29 88 67 55)(30 83 68 50)(31 86 69 53)(32 81 70 56)(33 89 74 43)(34 92 75 46)(35 95 76 41)(36 90 77 44)(37 93 78 47)(38 96 79 42)(39 91 80 45)(40 94 73 48)
(2 6)(4 8)(9 13)(11 15)(17 58)(18 63)(19 60)(20 57)(21 62)(22 59)(23 64)(24 61)(25 71)(26 68)(27 65)(28 70)(29 67)(30 72)(31 69)(32 66)(33 74)(34 79)(35 76)(36 73)(37 78)(38 75)(39 80)(40 77)(41 45)(43 47)(49 53)(51 55)(82 86)(84 88)(89 93)(91 95)```

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

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

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

Matrix representation of C3⋊C8.6D4 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 72 72 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 62 25 0 0 0 0 36 11 0 0 11 36 0 0 0 0 25 62 0 0
,
 34 3 0 0 0 0 28 39 0 0 0 0 0 0 62 37 0 0 0 0 48 11 0 0 0 0 0 0 11 48 0 0 0 0 37 62
,
 72 0 0 0 0 0 47 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,11,25,0,0,0,0,36,62,0,0,62,36,0,0,0,0,25,11,0,0],[34,28,0,0,0,0,3,39,0,0,0,0,0,0,62,48,0,0,0,0,37,11,0,0,0,0,0,0,11,37,0,0,0,0,48,62],[72,47,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;`

C3⋊C8.6D4 in GAP, Magma, Sage, TeX

`C_3\rtimes C_8._6D_4`
`% in TeX`

`G:=Group("C3:C8.6D4");`
`// GroupNames label`

`G:=SmallGroup(192,611);`
`// by ID`

`G=gap.SmallGroup(192,611);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,555,184,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^3,d*b*d=b^5,d*c*d=b^4*c^-1>;`
`// generators/relations`

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