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## G = C22.F9order 288 = 25·32

### The non-split extension by C22 of F9 acting via F9/C32⋊C4=C2

Aliases: C22.F9, C62.2C8, C322M5(2), C2.F92C2, C2.6(C2×F9), C3⋊Dic3.2C8, C322C8.6C4, C322C8.6C22, (C3×C6).6(C2×C8), (C2×C3⋊Dic3).5C4, C3⋊Dic3.4(C2×C4), (C2×C322C8).10C2, SmallGroup(288,866)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C22.F9
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C32⋊2C8 — C2.F9 — C22.F9
 Lower central C32 — C3×C6 — C22.F9
 Upper central C1 — C2 — C22

Generators and relations for C22.F9
G = < a,b,c,d,e | a2=b2=c3=d3=1, e8=b, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

Character table of C22.F9

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

Smallest permutation representation of C22.F9
On 48 points
Generators in S48
```(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(2 33 32)(3 34 17)(4 18 35)(6 20 37)(7 21 38)(8 39 22)(10 41 24)(11 42 25)(12 26 43)(14 28 45)(15 29 46)(16 47 30)
(1 48 31)(3 34 17)(4 35 18)(5 19 36)(7 21 38)(8 22 39)(9 40 23)(11 42 25)(12 43 26)(13 27 44)(15 29 46)(16 30 47)
(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)| (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,33,32)(3,34,17)(4,18,35)(6,20,37)(7,21,38)(8,39,22)(10,41,24)(11,42,25)(12,26,43)(14,28,45)(15,29,46)(16,47,30), (1,48,31)(3,34,17)(4,35,18)(5,19,36)(7,21,38)(8,22,39)(9,40,23)(11,42,25)(12,43,26)(13,27,44)(15,29,46)(16,30,47), (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( (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,33,32)(3,34,17)(4,18,35)(6,20,37)(7,21,38)(8,39,22)(10,41,24)(11,42,25)(12,26,43)(14,28,45)(15,29,46)(16,47,30), (1,48,31)(3,34,17)(4,35,18)(5,19,36)(7,21,38)(8,22,39)(9,40,23)(11,42,25)(12,43,26)(13,27,44)(15,29,46)(16,30,47), (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([(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(2,33,32),(3,34,17),(4,18,35),(6,20,37),(7,21,38),(8,39,22),(10,41,24),(11,42,25),(12,26,43),(14,28,45),(15,29,46),(16,47,30)], [(1,48,31),(3,34,17),(4,35,18),(5,19,36),(7,21,38),(8,22,39),(9,40,23),(11,42,25),(12,43,26),(13,27,44),(15,29,46),(16,30,47)], [(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 C22.F9 in GL10(𝔽97)

 1 0 0 0 0 0 0 0 0 0 0 96 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 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
,
 96 0 0 0 0 0 0 0 0 0 0 96 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 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
,
 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 50 50 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 47 0 1 0 0 0 0 0 0 50 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 47 0 1 0 0 0 0 0 0 50 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 47 0 1 0 0 0 0 0 0 0 47 0 0 1
,
 0 1 0 0 0 0 0 0 0 0 50 0 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 50 50 95 96 0 0 0 0 0 0 40 8 47 0 0 0 0 0 0 0 8 27 47 0 0 0 0 0

`G:=sub<GL(10,GF(97))| [1,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,50,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,96,96,47,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,96,96,47,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,96,47,47,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,50,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,50,40,8,0,0,0,0,0,0,0,50,8,27,0,0,0,0,0,0,96,95,47,47,0,0,0,0,0,0,1,96,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;`

C22.F9 in GAP, Magma, Sage, TeX

`C_2^2.F_9`
`% in TeX`

`G:=Group("C2^2.F9");`
`// GroupNames label`

`G:=SmallGroup(288,866);`
`// by ID`

`G=gap.SmallGroup(288,866);`
`# by ID`

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

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

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