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## G = C24.C4order 96 = 25·3

### 1st non-split extension by C24 of C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C24.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — C24.C4
 Lower central C3 — C6 — C12 — C24.C4
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for C24.C4
G = < a,b,c | a8=1, b6=a4, c2=a4b3, ab=ba, cac-1=a-1, cbc-1=b5 >

Character table of C24.C4

 class 1 2A 2B 3 4A 4B 4C 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 2 1 1 2 2 2 2 2 2 2 2 12 12 12 12 2 2 2 2 2 2 2 2 2 2 2 2 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 i i -i -i 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 linear of order 4 ρ6 1 1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 i -i -i i 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 linear of order 4 ρ7 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -i -i i i 1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 linear of order 4 ρ8 1 1 -1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -i i i -i 1 -1 1 -1 1 -1 -1 1 1 -1 -1 1 linear of order 4 ρ9 2 2 2 -1 2 2 2 -1 -1 -1 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 2 2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 -2 2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -1 2 2 -2 -1 1 1 0 0 0 0 0 0 0 0 1 -1 1 -1 -√3 √3 -√3 √3 -√3 -√3 √3 √3 orthogonal lifted from D12 ρ12 2 2 2 -1 2 2 2 -1 -1 -1 -2 -2 -2 -2 0 0 0 0 -1 -1 -1 -1 1 1 1 1 1 1 1 1 orthogonal lifted from D6 ρ13 2 2 -2 -1 2 2 -2 -1 1 1 0 0 0 0 0 0 0 0 1 -1 1 -1 √3 -√3 √3 -√3 √3 √3 -√3 -√3 orthogonal lifted from D12 ρ14 2 2 2 2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 2 -2 -1 -2 -2 2 -1 1 1 2 2 -2 -2 0 0 0 0 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 symplectic lifted from Dic3, Schur index 2 ρ16 2 2 -2 -1 -2 -2 2 -1 1 1 -2 -2 2 2 0 0 0 0 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 symplectic lifted from Dic3, Schur index 2 ρ17 2 2 2 -1 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 √3 √3 -√3 -√3 √3 -√3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ18 2 2 2 -1 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 0 0 1 1 1 1 -√3 -√3 √3 √3 -√3 √3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ19 2 -2 0 2 -2i 2i 0 -2 0 0 -√-2 √-2 -√2 √2 0 0 0 0 0 -2i 0 2i √2 √-2 √-2 √2 -√2 -√-2 -√-2 -√2 complex lifted from C8.C4 ρ20 2 -2 0 2 2i -2i 0 -2 0 0 -√-2 √-2 √2 -√2 0 0 0 0 0 2i 0 -2i -√2 √-2 √-2 -√2 √2 -√-2 -√-2 √2 complex lifted from C8.C4 ρ21 2 -2 0 2 -2i 2i 0 -2 0 0 √-2 -√-2 √2 -√2 0 0 0 0 0 -2i 0 2i -√2 -√-2 -√-2 -√2 √2 √-2 √-2 √2 complex lifted from C8.C4 ρ22 2 -2 0 2 2i -2i 0 -2 0 0 √-2 -√-2 -√2 √2 0 0 0 0 0 2i 0 -2i √2 -√-2 -√-2 √2 -√2 √-2 √-2 -√2 complex lifted from C8.C4 ρ23 2 -2 0 -1 2i -2i 0 1 -√-3 √-3 -√-2 √-2 √2 -√2 0 0 0 0 -√3 -i √3 i ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ32+ζ85ζ32+ζ85 complex faithful ρ24 2 -2 0 -1 -2i 2i 0 1 -√-3 √-3 √-2 -√-2 √2 -√2 0 0 0 0 √3 i -√3 -i ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ85ζ32+ζ85 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 complex faithful ρ25 2 -2 0 -1 -2i 2i 0 1 √-3 -√-3 √-2 -√-2 √2 -√2 0 0 0 0 -√3 i √3 -i ζ83ζ3+ζ8ζ3+ζ8 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ32+ζ85ζ32+ζ85 complex faithful ρ26 2 -2 0 -1 2i -2i 0 1 √-3 -√-3 √-2 -√-2 -√2 √2 0 0 0 0 √3 -i -√3 i ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ83ζ3+ζ8ζ3+ζ8 complex faithful ρ27 2 -2 0 -1 2i -2i 0 1 -√-3 √-3 √-2 -√-2 -√2 √2 0 0 0 0 -√3 -i √3 i ζ83ζ32+ζ83+ζ8ζ32 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87+ζ85ζ3 complex faithful ρ28 2 -2 0 -1 -2i 2i 0 1 √-3 -√-3 -√-2 √-2 -√2 √2 0 0 0 0 -√3 i √3 -i ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ32+ζ87-ζ85ζ32 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ3+ζ8ζ3+ζ8 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 complex faithful ρ29 2 -2 0 -1 -2i 2i 0 1 -√-3 √-3 -√-2 √-2 -√2 √2 0 0 0 0 √3 i -√3 -i ζ87ζ32+ζ85ζ32+ζ85 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ32+ζ83+ζ8ζ32 ζ87ζ3+ζ87+ζ85ζ3 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ3+ζ8ζ3+ζ8 complex faithful ρ30 2 -2 0 -1 2i -2i 0 1 √-3 -√-3 -√-2 √-2 √2 -√2 0 0 0 0 √3 -i -√3 i ζ87ζ3+ζ87+ζ85ζ3 ζ87ζ3+ζ87-ζ85ζ3 ζ87ζ32+ζ87-ζ85ζ32 ζ83ζ3+ζ8ζ3+ζ8 ζ87ζ32+ζ85ζ32+ζ85 ζ83ζ32+ζ83-ζ8ζ32 ζ83ζ3+ζ83-ζ8ζ3 ζ83ζ32+ζ83+ζ8ζ32 complex faithful

Smallest permutation representation of C24.C4
On 48 points
Generators in S48
```(1 20 10 17 7 14 4 23)(2 21 11 18 8 15 5 24)(3 22 12 19 9 16 6 13)(25 46 28 37 31 40 34 43)(26 47 29 38 32 41 35 44)(27 48 30 39 33 42 36 45)
(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)
(1 25 10 34 7 31 4 28)(2 30 11 27 8 36 5 33)(3 35 12 32 9 29 6 26)(13 44 22 41 19 38 16 47)(14 37 23 46 20 43 17 40)(15 42 24 39 21 48 18 45)```

`G:=sub<Sym(48)| (1,20,10,17,7,14,4,23)(2,21,11,18,8,15,5,24)(3,22,12,19,9,16,6,13)(25,46,28,37,31,40,34,43)(26,47,29,38,32,41,35,44)(27,48,30,39,33,42,36,45), (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), (1,25,10,34,7,31,4,28)(2,30,11,27,8,36,5,33)(3,35,12,32,9,29,6,26)(13,44,22,41,19,38,16,47)(14,37,23,46,20,43,17,40)(15,42,24,39,21,48,18,45)>;`

`G:=Group( (1,20,10,17,7,14,4,23)(2,21,11,18,8,15,5,24)(3,22,12,19,9,16,6,13)(25,46,28,37,31,40,34,43)(26,47,29,38,32,41,35,44)(27,48,30,39,33,42,36,45), (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), (1,25,10,34,7,31,4,28)(2,30,11,27,8,36,5,33)(3,35,12,32,9,29,6,26)(13,44,22,41,19,38,16,47)(14,37,23,46,20,43,17,40)(15,42,24,39,21,48,18,45) );`

`G=PermutationGroup([(1,20,10,17,7,14,4,23),(2,21,11,18,8,15,5,24),(3,22,12,19,9,16,6,13),(25,46,28,37,31,40,34,43),(26,47,29,38,32,41,35,44),(27,48,30,39,33,42,36,45)], [(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)], [(1,25,10,34,7,31,4,28),(2,30,11,27,8,36,5,33),(3,35,12,32,9,29,6,26),(13,44,22,41,19,38,16,47),(14,37,23,46,20,43,17,40),(15,42,24,39,21,48,18,45)])`

Matrix representation of C24.C4 in GL2(𝔽73) generated by

 63 0 28 51
,
 49 0 24 70
,
 72 63 12 1
`G:=sub<GL(2,GF(73))| [63,28,0,51],[49,24,0,70],[72,12,63,1] >;`

C24.C4 in GAP, Magma, Sage, TeX

`C_{24}.C_4`
`% in TeX`

`G:=Group("C24.C4");`
`// GroupNames label`

`G:=SmallGroup(96,26);`
`// by ID`

`G=gap.SmallGroup(96,26);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,121,55,86,579,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^6=a^4,c^2=a^4*b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^5>;`
`// generators/relations`

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