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## G = C24.8D4order 192 = 26·3

### 8th non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24.8D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C2×Dic12 — C24.8D4
 Lower central C3 — C6 — C12 — C24 — C24.8D4
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for C24.8D4
G = < a,b,c | a24=1, b4=a12, c2=a21, bab-1=a7, cac-1=a17, cbc-1=a21b3 >

Character table of C24.8D4

 class 1 2A 2B 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 12A 12B 12C 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 2 2 2 24 24 2 4 2 2 4 8 8 2 2 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 -i i -1 -1 1 -i i -i i -1 1 -1 1 i -i -i i linear of order 4 ρ6 1 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 -i i -1 -1 1 i -i i -i -1 1 -1 1 i -i -i i linear of order 4 ρ7 1 1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 i -i -1 -1 1 -i i -i i -1 1 -1 1 -i i i -i linear of order 4 ρ8 1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 1 i -i -1 -1 1 i -i i -i -1 1 -1 1 -i i i -i linear of order 4 ρ9 2 2 2 2 2 2 0 0 2 2 -2 -2 -2 0 0 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -1 2 2 0 0 -1 -1 2 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 -2 2 -2 2 0 0 2 -2 2 2 -2 0 0 -2 -2 2 0 0 0 0 2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -1 2 2 0 0 -1 -1 2 2 2 -2 -2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 -1 -2 2 0 0 -1 1 2 2 -2 0 0 1 1 -1 0 0 0 0 -1 1 -1 1 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ14 2 2 -2 -1 -2 2 0 0 -1 1 2 2 -2 0 0 1 1 -1 0 0 0 0 -1 1 -1 1 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ15 2 2 -2 2 2 -2 0 0 2 -2 0 0 0 0 0 2 2 -2 -√2 -√2 √2 √2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ16 2 2 -2 2 2 -2 0 0 2 -2 0 0 0 0 0 2 2 -2 √2 √2 -√2 -√2 0 0 0 0 0 0 0 0 orthogonal lifted from D8 ρ17 2 2 2 -1 2 2 0 0 -1 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 2 -1 2 2 0 0 -1 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ19 2 2 -2 -1 -2 2 0 0 -1 1 -2 -2 2 2i -2i 1 1 -1 0 0 0 0 1 -1 1 -1 i -i -i i complex lifted from C4×S3 ρ20 2 2 -2 -1 -2 2 0 0 -1 1 -2 -2 2 -2i 2i 1 1 -1 0 0 0 0 1 -1 1 -1 -i i i -i complex lifted from C4×S3 ρ21 2 2 2 2 -2 -2 0 0 2 2 0 0 0 0 0 -2 -2 -2 -√-2 √-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 -2 -2 0 0 2 2 0 0 0 0 0 -2 -2 -2 √-2 -√-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ23 4 4 -4 -2 4 -4 0 0 -2 2 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ24 4 4 4 -2 -4 -4 0 0 -2 -2 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ25 4 -4 0 4 0 0 0 0 -4 0 -2√2 2√2 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 0 0 symplectic lifted from C8.17D4, Schur index 2 ρ26 4 -4 0 4 0 0 0 0 -4 0 2√2 -2√2 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 0 0 symplectic lifted from C8.17D4, Schur index 2 ρ27 4 -4 0 -2 0 0 0 0 2 0 2√2 -2√2 0 0 0 2√3 -2√3 0 0 0 0 0 √2 -√6 -√2 √6 0 0 0 0 symplectic faithful, Schur index 2 ρ28 4 -4 0 -2 0 0 0 0 2 0 -2√2 2√2 0 0 0 -2√3 2√3 0 0 0 0 0 -√2 -√6 √2 √6 0 0 0 0 symplectic faithful, Schur index 2 ρ29 4 -4 0 -2 0 0 0 0 2 0 2√2 -2√2 0 0 0 -2√3 2√3 0 0 0 0 0 √2 √6 -√2 -√6 0 0 0 0 symplectic faithful, Schur index 2 ρ30 4 -4 0 -2 0 0 0 0 2 0 -2√2 2√2 0 0 0 2√3 -2√3 0 0 0 0 0 -√2 √6 √2 -√6 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C24.8D4
On 96 points
Generators in S96
```(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 40 7 34 13 28 19 46)(2 47 8 41 14 35 20 29)(3 30 9 48 15 42 21 36)(4 37 10 31 16 25 22 43)(5 44 11 38 17 32 23 26)(6 27 12 45 18 39 24 33)(49 92 55 86 61 80 67 74)(50 75 56 93 62 87 68 81)(51 82 57 76 63 94 69 88)(52 89 58 83 64 77 70 95)(53 96 59 90 65 84 71 78)(54 79 60 73 66 91 72 85)
(1 83 22 80 19 77 16 74 13 95 10 92 7 89 4 86)(2 76 23 73 20 94 17 91 14 88 11 85 8 82 5 79)(3 93 24 90 21 87 18 84 15 81 12 78 9 75 6 96)(25 52 46 49 43 70 40 67 37 64 34 61 31 58 28 55)(26 69 47 66 44 63 41 60 38 57 35 54 32 51 29 72)(27 62 48 59 45 56 42 53 39 50 36 71 33 68 30 65)```

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

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

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

Matrix representation of C24.8D4 in GL6(𝔽97)

 1 1 0 0 0 0 96 0 0 0 0 0 0 0 14 83 0 0 0 0 7 0 0 0 0 0 0 0 0 14 0 0 0 0 90 14
,
 75 0 0 0 0 0 0 75 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 96 2 0 0 0 0 96 1 0 0
,
 88 81 0 0 0 0 90 9 0 0 0 0 0 0 0 0 54 77 0 0 0 0 44 43 0 0 34 20 0 0 0 0 44 63 0 0

`G:=sub<GL(6,GF(97))| [1,96,0,0,0,0,1,0,0,0,0,0,0,0,14,7,0,0,0,0,83,0,0,0,0,0,0,0,0,90,0,0,0,0,14,14],[75,0,0,0,0,0,0,75,0,0,0,0,0,0,0,0,96,96,0,0,0,0,2,1,0,0,1,0,0,0,0,0,0,1,0,0],[88,90,0,0,0,0,81,9,0,0,0,0,0,0,0,0,34,44,0,0,0,0,20,63,0,0,54,44,0,0,0,0,77,43,0,0] >;`

C24.8D4 in GAP, Magma, Sage, TeX

`C_{24}._8D_4`
`% in TeX`

`G:=Group("C24.8D4");`
`// GroupNames label`

`G:=SmallGroup(192,55);`
`// by ID`

`G=gap.SmallGroup(192,55);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,141,36,758,184,675,794,192,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^21,b*a*b^-1=a^7,c*a*c^-1=a^17,c*b*c^-1=a^21*b^3>;`
`// generators/relations`

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