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## G = C4.F9order 288 = 25·32

### The non-split extension by C4 of F9 acting via F9/C32⋊C4=C2

Aliases: C4.F9, C321M5(2), C2.F91C2, C2.4(C2×F9), (C3×C12).3C8, C322C8.5C4, C322C8.4C22, (C4×C3⋊S3).2C4, (C2×C3⋊S3).2C8, (C3×C6).2(C2×C8), C3⋊S33C8.9C2, C3⋊Dic3.2(C2×C4), SmallGroup(288,862)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C4.F9
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2.F9 — C4.F9
 Lower central C32 — C3×C6 — C4.F9
 Upper central C1 — C2 — C4

Generators and relations for C4.F9
G = < a,b,c,d | a4=b3=c3=1, d8=a2, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >

Character table of C4.F9

 class 1 2A 2B 3 4A 4B 4C 6 8A 8B 8C 8D 8E 8F 12A 12B 16A 16B 16C 16D 16E 16F 16G 16H size 1 1 18 8 2 9 9 8 9 9 9 9 18 18 8 8 18 18 18 18 18 18 18 18 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 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 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 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 linear of order 2 ρ5 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -i -i i i i i -i -i linear of order 4 ρ6 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 i i -i -i -i -i i i linear of order 4 ρ7 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -i -i i i -i -i i i linear of order 4 ρ8 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 i i -i -i i i -i -i linear of order 4 ρ9 1 1 1 1 -1 -1 -1 1 -i -i i i i -i -1 -1 ζ8 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 ζ83 linear of order 8 ρ10 1 1 1 1 -1 -1 -1 1 i i -i -i -i i -1 -1 ζ87 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 ζ85 linear of order 8 ρ11 1 1 1 1 -1 -1 -1 1 -i -i i i i -i -1 -1 ζ85 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 ζ87 linear of order 8 ρ12 1 1 1 1 -1 -1 -1 1 i i -i -i -i i -1 -1 ζ83 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 ζ8 linear of order 8 ρ13 1 1 -1 1 1 -1 -1 1 -i -i i i -i i 1 1 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 linear of order 8 ρ14 1 1 -1 1 1 -1 -1 1 i i -i -i i -i 1 1 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 linear of order 8 ρ15 1 1 -1 1 1 -1 -1 1 i i -i -i i -i 1 1 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 linear of order 8 ρ16 1 1 -1 1 1 -1 -1 1 -i -i i i -i i 1 1 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 linear of order 8 ρ17 2 -2 0 2 0 2i -2i -2 2ζ8 2ζ85 2ζ83 2ζ87 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M5(2) ρ18 2 -2 0 2 0 -2i 2i -2 2ζ87 2ζ83 2ζ85 2ζ8 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M5(2) ρ19 2 -2 0 2 0 2i -2i -2 2ζ85 2ζ8 2ζ87 2ζ83 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M5(2) ρ20 2 -2 0 2 0 -2i 2i -2 2ζ83 2ζ87 2ζ8 2ζ85 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from M5(2) ρ21 8 8 0 -1 8 0 0 -1 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 orthogonal lifted from F9 ρ22 8 8 0 -1 -8 0 0 -1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C2×F9 ρ23 8 -8 0 -1 0 0 0 1 0 0 0 0 0 0 -3i 3i 0 0 0 0 0 0 0 0 complex faithful ρ24 8 -8 0 -1 0 0 0 1 0 0 0 0 0 0 3i -3i 0 0 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C4.F9
On 48 points
Generators in S48
```(1 13 9 5)(2 6 10 14)(3 15 11 7)(4 8 12 16)(17 45 25 37)(18 38 26 46)(19 47 27 39)(20 40 28 48)(21 33 29 41)(22 42 30 34)(23 35 31 43)(24 44 32 36)
(2 25 33)(3 26 34)(4 35 27)(6 37 29)(7 38 30)(8 31 39)(10 17 41)(11 18 42)(12 43 19)(14 45 21)(15 46 22)(16 23 47)
(1 24 48)(3 26 34)(4 27 35)(5 36 28)(7 38 30)(8 39 31)(9 32 40)(11 18 42)(12 19 43)(13 44 20)(15 46 22)(16 47 23)
(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)```

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

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

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

Matrix representation of C4.F9 in GL10(𝔽97)

 22 0 0 0 0 0 0 0 0 0 75 75 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 0 96
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 96 96 0 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 0 96 0 1 0 0 0 0 0 0 1 0 96 96
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 1 96 0 0 0 0 0 0 0 0 0 96 0 1 0 0 0 0 0 0 1 0 96 96 0 0 0 0 0 0 0 0 0 0 96 1 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 96 0 1 0 0 0 0 0 0 0 96 0 0 1
,
 96 95 0 0 0 0 0 0 0 0 74 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 96 1 0 0 0 0 0 0 1 1 95 96 0 0 0 0 0 0 33 33 96 0 0 0 0 0 0 0 33 32 96 0 0 0 0 0

`G:=sub<GL(10,GF(97))| [22,75,0,0,0,0,0,0,0,0,0,75,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,96,96,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,96,96,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,96,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[96,74,0,0,0,0,0,0,0,0,95,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,33,33,0,0,0,0,0,0,0,1,33,32,0,0,0,0,0,0,96,95,96,96,0,0,0,0,0,0,1,96,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C4.F9 in GAP, Magma, Sage, TeX

`C_4.F_9`
`% in TeX`

`G:=Group("C4.F9");`
`// GroupNames label`

`G:=SmallGroup(288,862);`
`// by ID`

`G=gap.SmallGroup(288,862);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,120,58,80,4037,2371,362,10982,3156,1203]);`
`// Polycyclic`

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

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