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## G = C6.(C4○D8)  order 192 = 26·3

### 22nd non-split extension by C6 of C4○D8 acting via C4○D8/SD16=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C6.(C4○D8)
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C4⋊C4⋊7S3 — C6.(C4○D8)
 Lower central C3 — C6 — C2×C12 — C6.(C4○D8)
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C6.(C4○D8)
G = < a,b,c,d | a6=b4=1, c4=b2, d2=a3, bab-1=cac-1=dad-1=a-1, cbc-1=a3b, bd=db, dcd-1=b2c3 >

Subgroups: 272 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C23.20D4, C6.Q16, C6.SD16, C2.Dic12, D6⋊C8, C3×C4.Q8, C4⋊C47S3, C4.D12, C6.(C4○D8)
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D8, C8.C22, C4○D12, S3×D4, Q83S3, C23.20D4, D6.D4, D4.D6, Q8.7D6, C6.(C4○D8)

Smallest permutation representation of C6.(C4○D8)
On 96 points
Generators in S96
```(1 71 61 83 42 10)(2 11 43 84 62 72)(3 65 63 85 44 12)(4 13 45 86 64 66)(5 67 57 87 46 14)(6 15 47 88 58 68)(7 69 59 81 48 16)(8 9 41 82 60 70)(17 93 75 51 27 38)(18 39 28 52 76 94)(19 95 77 53 29 40)(20 33 30 54 78 96)(21 89 79 55 31 34)(22 35 32 56 80 90)(23 91 73 49 25 36)(24 37 26 50 74 92)
(1 85 5 81)(2 4 6 8)(3 87 7 83)(9 72 13 68)(10 44 14 48)(11 66 15 70)(12 46 16 42)(17 23 21 19)(18 50 22 54)(20 52 24 56)(25 79 29 75)(26 35 30 39)(27 73 31 77)(28 37 32 33)(34 95 38 91)(36 89 40 93)(41 62 45 58)(43 64 47 60)(49 55 53 51)(57 69 61 65)(59 71 63 67)(74 90 78 94)(76 92 80 96)(82 84 86 88)
(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 18 83 52)(2 17 84 51)(3 24 85 50)(4 23 86 49)(5 22 87 56)(6 21 88 55)(7 20 81 54)(8 19 82 53)(9 40 60 77)(10 39 61 76)(11 38 62 75)(12 37 63 74)(13 36 64 73)(14 35 57 80)(15 34 58 79)(16 33 59 78)(25 66 91 45)(26 65 92 44)(27 72 93 43)(28 71 94 42)(29 70 95 41)(30 69 96 48)(31 68 89 47)(32 67 90 46)```

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

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

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

33 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 12 2 2 2 4 4 6 6 8 12 12 24 2 2 2 4 4 12 12 4 4 8 8 8 8 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C4○D8 C4○D12 C8.C22 Q8⋊3S3 S3×D4 D4.D6 Q8.7D6 kernel C6.(C4○D8) C6.Q16 C6.SD16 C2.Dic12 D6⋊C8 C3×C4.Q8 C4⋊C4⋊7S3 C4.D12 C4.Q8 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C12 C6 C4 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 1 2 1 4 4 4 1 1 1 2 2

Matrix representation of C6.(C4○D8) in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 72 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 72 0 0 0 0 72 0 0 0 0 0 0 0 22 0 0 0 0 0 55 10 0 0 0 0 0 0 0 72 0 0 0 0 72 0
,
 46 0 0 0 0 0 0 46 0 0 0 0 0 0 70 71 0 0 0 0 4 3 0 0 0 0 0 0 0 72 0 0 0 0 72 0

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,22,55,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,72,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,70,4,0,0,0,0,71,3,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;`

C6.(C4○D8) in GAP, Magma, Sage, TeX

`C_6.(C_4\circ D_8)`
`% in TeX`

`G:=Group("C6.(C4oD8)");`
`// GroupNames label`

`G:=SmallGroup(192,427);`
`// by ID`

`G=gap.SmallGroup(192,427);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,926,219,100,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^4=1,c^4=b^2,d^2=a^3,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=b^2*c^3>;`
`// generators/relations`

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