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## G = C5⋊F9order 360 = 23·32·5

### The semidirect product of C5 and F9 acting via F9/C3⋊S3=C4

Aliases: C5⋊F9, C32⋊(C5⋊C8), C3⋊S3.F5, (C3×C15)⋊3C8, C32⋊Dic5.1C2, (C5×C3⋊S3).3C4, SmallGroup(360,125)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C5⋊F9
 Chief series C1 — C5 — C3×C15 — C5×C3⋊S3 — C32⋊Dic5 — C5⋊F9
 Lower central C3×C15 — C5⋊F9
 Upper central C1

Generators and relations for C5⋊F9
G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a3, dbd-1=bc=cb, dcd-1=b >

Character table of C5⋊F9

 class 1 2 3 4A 4B 5 8A 8B 8C 8D 10 15A 15B 15C 15D size 1 9 8 45 45 4 45 45 45 45 36 8 8 8 8 ρ1 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 linear of order 2 ρ3 1 1 1 -1 -1 1 i -i i -i 1 1 1 1 1 linear of order 4 ρ4 1 1 1 -1 -1 1 -i i -i i 1 1 1 1 1 linear of order 4 ρ5 1 -1 1 -i i 1 ζ8 ζ83 ζ85 ζ87 -1 1 1 1 1 linear of order 8 ρ6 1 -1 1 i -i 1 ζ83 ζ8 ζ87 ζ85 -1 1 1 1 1 linear of order 8 ρ7 1 -1 1 -i i 1 ζ85 ζ87 ζ8 ζ83 -1 1 1 1 1 linear of order 8 ρ8 1 -1 1 i -i 1 ζ87 ζ85 ζ83 ζ8 -1 1 1 1 1 linear of order 8 ρ9 4 4 4 0 0 -1 0 0 0 0 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ10 4 -4 4 0 0 -1 0 0 0 0 1 -1 -1 -1 -1 symplectic lifted from C5⋊C8, Schur index 2 ρ11 8 0 -1 0 0 8 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F9 ρ12 8 0 -1 0 0 -2 0 0 0 0 0 3ζ5+1 3ζ54+1 3ζ52+1 3ζ53+1 complex faithful ρ13 8 0 -1 0 0 -2 0 0 0 0 0 3ζ52+1 3ζ53+1 3ζ54+1 3ζ5+1 complex faithful ρ14 8 0 -1 0 0 -2 0 0 0 0 0 3ζ53+1 3ζ52+1 3ζ5+1 3ζ54+1 complex faithful ρ15 8 0 -1 0 0 -2 0 0 0 0 0 3ζ54+1 3ζ5+1 3ζ53+1 3ζ52+1 complex faithful

Smallest permutation representation of C5⋊F9
On 45 points
Generators in S45
```(1 4 5 3 2)(6 35 39 24 20)(7 25 36 21 40)(8 14 26 41 37)(9 42 15 30 27)(10 31 43 28 16)(11 29 32 17 44)(12 18 22 45 33)(13 38 19 34 23)
(1 29 25)(2 11 7)(3 44 40)(4 32 36)(5 17 21)(6 37 38)(8 19 35)(9 45 16)(10 42 33)(12 31 15)(13 20 41)(14 34 39)(18 43 30)(22 28 27)(23 24 26)
(1 22 26)(2 18 14)(3 12 8)(4 45 41)(5 33 37)(6 21 42)(7 30 39)(9 20 36)(10 38 17)(11 43 34)(13 32 16)(15 35 40)(19 44 31)(23 29 28)(24 25 27)
(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)```

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

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

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

Matrix representation of C5⋊F9 in GL8(𝔽241)

 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 240 240 240 240 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 143 179 92 1 98 62 149 0 24 111 106 129 196 64 133 91
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 143 179 92 1 98 62 149 0 186 197 213 94 240 0 0 0 147 92 103 119 0 240 0 0 122 28 214 225 0 0 240 0 82 93 213 229 175 70 182 1
,
 27 94 120 154 111 72 112 1 221 29 228 207 86 130 23 91 205 13 16 215 87 106 236 87 194 76 125 174 174 43 87 205 0 0 0 0 0 0 0 1 24 111 106 129 196 64 133 91 107 63 174 40 22 47 172 87 148 192 166 28 66 171 59 240
,
 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 240 240 240 240 0 0 0 0 0 0 0 0 0 0 0 1 134 24 5 202 168 107 132 205 24 111 106 129 196 64 133 91 82 93 213 229 175 70 182 1

`G:=sub<GL(8,GF(241))| [0,0,0,240,0,0,143,24,1,0,0,240,0,0,179,111,0,1,0,240,0,0,92,106,0,0,1,240,0,0,1,129,0,0,0,0,0,0,98,196,0,0,0,0,1,0,62,64,0,0,0,0,0,1,149,133,0,0,0,0,0,0,0,91],[0,0,0,143,186,147,122,82,0,0,0,179,197,92,28,93,0,0,0,92,213,103,214,213,0,0,0,1,94,119,225,229,1,0,0,98,240,0,0,175,0,1,0,62,0,240,0,70,0,0,1,149,0,0,240,182,0,0,0,0,0,0,0,1],[27,221,205,194,0,24,107,148,94,29,13,76,0,111,63,192,120,228,16,125,0,106,174,166,154,207,215,174,0,129,40,28,111,86,87,174,0,196,22,66,72,130,106,43,0,64,47,171,112,23,236,87,0,133,172,59,1,91,87,205,1,91,87,240],[1,0,0,240,0,134,24,82,0,0,1,240,0,24,111,93,0,0,0,240,0,5,106,213,0,1,0,240,0,202,129,229,0,0,0,0,0,168,196,175,0,0,0,0,0,107,64,70,0,0,0,0,0,132,133,182,0,0,0,0,1,205,91,1] >;`

C5⋊F9 in GAP, Magma, Sage, TeX

`C_5\rtimes F_9`
`% in TeX`

`G:=Group("C5:F9");`
`// GroupNames label`

`G:=SmallGroup(360,125);`
`// by ID`

`G=gap.SmallGroup(360,125);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,3,-5,12,31,1155,681,111,1204,970,376,5189,5195]);`
`// Polycyclic`

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

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