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## G = C8.2D12order 192 = 26·3

### 2nd non-split extension by C8 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C8.2D12
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C2×C8⋊S3 — C8.2D12
 Lower central C3 — C6 — C2×C12 — C8.2D12
 Upper central C1 — C22 — C2×C4 — C4.Q8

Generators and relations for C8.2D12
G = < a,b,c | a8=b12=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >

Subgroups: 320 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C8⋊S3, Dic12, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C8.D4, C6.SD16, C3×C4.Q8, C4.D12, C2×C8⋊S3, C2×Dic12, C8.2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C8.C22, C2×D12, S3×D4, Q83S3, C8.D4, C12⋊D4, D4.D6, C8.2D12

Character table of C8.2D12

 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 24A 24B 24C 24D size 1 1 1 1 12 2 2 2 8 8 12 24 24 2 2 2 4 4 12 12 4 4 8 8 8 8 4 4 4 4 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 -2 2 0 0 2 -2 0 0 0 0 -2 2 2 -2 orthogonal lifted from D4 ρ10 2 -2 -2 2 0 2 -2 2 0 0 0 0 0 -2 -2 2 2 -2 0 0 2 -2 0 0 0 0 2 -2 -2 2 orthogonal lifted from D4 ρ11 2 2 2 2 0 -1 2 2 2 2 0 0 0 -1 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 0 -1 2 2 -2 2 0 0 0 -1 -1 -1 -2 -2 0 0 -1 -1 -1 -1 1 1 1 1 1 1 orthogonal lifted from D6 ρ13 2 2 2 2 2 2 -2 -2 0 0 -2 0 0 2 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 2 -2 -2 0 0 2 0 0 2 2 2 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 0 -1 2 2 -2 -2 0 0 0 -1 -1 -1 2 2 0 0 -1 -1 1 1 1 1 -1 -1 -1 -1 orthogonal lifted from D6 ρ16 2 2 2 2 0 -1 2 2 2 -2 0 0 0 -1 -1 -1 -2 -2 0 0 -1 -1 1 1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ17 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 1 1 -1 2 -2 0 0 -1 1 -√3 √3 √3 -√3 -1 1 1 -1 orthogonal lifted from D12 ρ18 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 1 1 -1 2 -2 0 0 -1 1 √3 -√3 -√3 √3 -1 1 1 -1 orthogonal lifted from D12 ρ19 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 1 1 -1 -2 2 0 0 -1 1 √3 -√3 √3 -√3 1 -1 -1 1 orthogonal lifted from D12 ρ20 2 -2 -2 2 0 -1 -2 2 0 0 0 0 0 1 1 -1 -2 2 0 0 -1 1 -√3 √3 -√3 √3 1 -1 -1 1 orthogonal lifted from D12 ρ21 2 -2 -2 2 0 2 2 -2 0 0 0 0 0 -2 -2 2 0 0 2i -2i -2 2 0 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 0 0 -2i 2i -2 2 0 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 2 2 0 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 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, 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 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 0 0 0 -√6 √6 -√6 √6 symplectic lifted from D4.D6, 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 0 0 0 √6 √6 -√6 -√6 symplectic lifted from D4.D6, 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 -√6 -√6 √6 √6 symplectic lifted from D4.D6, 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 √6 -√6 √6 -√6 symplectic lifted from D4.D6, Schur index 2

Smallest permutation representation of C8.2D12
On 96 points
Generators in S96
```(1 40 65 16 57 92 31 73)(2 17 32 41 58 74 66 93)(3 42 67 18 59 94 33 75)(4 19 34 43 60 76 68 95)(5 44 69 20 49 96 35 77)(6 21 36 45 50 78 70 85)(7 46 71 22 51 86 25 79)(8 23 26 47 52 80 72 87)(9 48 61 24 53 88 27 81)(10 13 28 37 54 82 62 89)(11 38 63 14 55 90 29 83)(12 15 30 39 56 84 64 91)
(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 56 57 12)(2 11 58 55)(3 54 59 10)(4 9 60 53)(5 52 49 8)(6 7 50 51)(13 75 82 18)(14 17 83 74)(15 73 84 16)(19 81 76 24)(20 23 77 80)(21 79 78 22)(25 70 71 36)(26 35 72 69)(27 68 61 34)(28 33 62 67)(29 66 63 32)(30 31 64 65)(37 94 89 42)(38 41 90 93)(39 92 91 40)(43 88 95 48)(44 47 96 87)(45 86 85 46)```

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

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

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

Matrix representation of C8.2D12 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 34 5 39 68 0 0 68 39 5 34 0 0 34 5 34 5 0 0 68 39 68 39
,
 72 71 0 0 0 0 1 1 0 0 0 0 0 0 43 15 27 23 0 0 58 58 50 50 0 0 27 23 30 58 0 0 50 50 15 15
,
 72 71 0 0 0 0 0 1 0 0 0 0 0 0 58 58 50 50 0 0 43 15 27 23 0 0 50 50 15 15 0 0 27 23 30 58

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,34,68,34,68,0,0,5,39,5,39,0,0,39,5,34,68,0,0,68,34,5,39],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,43,58,27,50,0,0,15,58,23,50,0,0,27,50,30,15,0,0,23,50,58,15],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,58,43,50,27,0,0,58,15,50,23,0,0,50,27,15,30,0,0,50,23,15,58] >;`

C8.2D12 in GAP, Magma, Sage, TeX

`C_8._2D_{12}`
`% in TeX`

`G:=Group("C8.2D12");`
`// GroupNames label`

`G:=SmallGroup(192,426);`
`// by ID`

`G=gap.SmallGroup(192,426);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,254,555,226,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=b^12=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;`
`// generators/relations`

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