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## G = C9⋊F5order 180 = 22·32·5

### The semidirect product of C9 and F5 acting via F5/D5=C2

Aliases: C9⋊F5, C5⋊Dic9, C451C4, D5.D9, C15.Dic3, C3.(C3⋊F5), (C9×D5).1C2, (C3×D5).2S3, SmallGroup(180,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — C9⋊F5
 Chief series C1 — C3 — C15 — C45 — C9×D5 — C9⋊F5
 Lower central C45 — C9⋊F5
 Upper central C1

Generators and relations for C9⋊F5
G = < a,b,c | a9=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >

Character table of C9⋊F5

 class 1 2 3 4A 4B 5 6 9A 9B 9C 15A 15B 18A 18B 18C 45A 45B 45C 45D 45E 45F size 1 5 2 45 45 4 10 2 2 2 4 4 10 10 10 4 4 4 4 4 4 ρ1 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 linear of order 2 ρ3 1 -1 1 -i i 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 linear of order 4 ρ4 1 -1 1 i -i 1 -1 1 1 1 1 1 -1 -1 -1 1 1 1 1 1 1 linear of order 4 ρ5 2 2 2 0 0 2 2 -1 -1 -1 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 -1 0 0 2 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ7 2 2 -1 0 0 2 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ8 2 2 -1 0 0 2 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ9 2 -2 -1 0 0 2 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -1 -1 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 symplectic lifted from Dic9, Schur index 2 ρ10 2 -2 -1 0 0 2 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -1 -1 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 symplectic lifted from Dic9, Schur index 2 ρ11 2 -2 2 0 0 2 -2 -1 -1 -1 2 2 1 1 1 -1 -1 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 -1 0 0 2 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -1 -1 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 symplectic lifted from Dic9, Schur index 2 ρ13 4 0 4 0 0 -1 0 4 4 4 -1 -1 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ14 4 0 4 0 0 -1 0 -2 -2 -2 -1 -1 0 0 0 1-√-15/2 1-√-15/2 1+√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ15 4 0 4 0 0 -1 0 -2 -2 -2 -1 -1 0 0 0 1+√-15/2 1+√-15/2 1-√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ16 4 0 -2 0 0 -1 0 2ζ95+2ζ94 2ζ98+2ζ9 2ζ97+2ζ92 1+√-15/2 1-√-15/2 0 0 0 -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 complex faithful ρ17 4 0 -2 0 0 -1 0 2ζ95+2ζ94 2ζ98+2ζ9 2ζ97+2ζ92 1-√-15/2 1+√-15/2 0 0 0 -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 complex faithful ρ18 4 0 -2 0 0 -1 0 2ζ97+2ζ92 2ζ95+2ζ94 2ζ98+2ζ9 1+√-15/2 1-√-15/2 0 0 0 ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 complex faithful ρ19 4 0 -2 0 0 -1 0 2ζ98+2ζ9 2ζ97+2ζ92 2ζ95+2ζ94 1+√-15/2 1-√-15/2 0 0 0 -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 complex faithful ρ20 4 0 -2 0 0 -1 0 2ζ97+2ζ92 2ζ95+2ζ94 2ζ98+2ζ9 1-√-15/2 1+√-15/2 0 0 0 -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 complex faithful ρ21 4 0 -2 0 0 -1 0 2ζ98+2ζ9 2ζ97+2ζ92 2ζ95+2ζ94 1-√-15/2 1+√-15/2 0 0 0 ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 complex faithful

Smallest permutation representation of C9⋊F5
On 45 points
Generators in S45
```(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)
(1 10 24 43 33)(2 11 25 44 34)(3 12 26 45 35)(4 13 27 37 36)(5 14 19 38 28)(6 15 20 39 29)(7 16 21 40 30)(8 17 22 41 31)(9 18 23 42 32)
(2 9)(3 8)(4 7)(5 6)(10 24 33 43)(11 23 34 42)(12 22 35 41)(13 21 36 40)(14 20 28 39)(15 19 29 38)(16 27 30 37)(17 26 31 45)(18 25 32 44)```

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

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

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

C9⋊F5 is a maximal subgroup of   D9×F5
C9⋊F5 is a maximal quotient of   C45⋊C8

Matrix representation of C9⋊F5 in GL4(𝔽181) generated by

 4 54 0 0 127 131 0 0 0 0 4 54 0 0 127 131
,
 74 149 180 0 32 106 0 180 1 0 0 0 0 1 0 0
,
 1 0 0 0 180 180 0 0 107 32 107 32 106 74 106 74
`G:=sub<GL(4,GF(181))| [4,127,0,0,54,131,0,0,0,0,4,127,0,0,54,131],[74,32,1,0,149,106,0,1,180,0,0,0,0,180,0,0],[1,180,107,106,0,180,32,74,0,0,107,106,0,0,32,74] >;`

C9⋊F5 in GAP, Magma, Sage, TeX

`C_9\rtimes F_5`
`% in TeX`

`G:=Group("C9:F5");`
`// GroupNames label`

`G:=SmallGroup(180,6);`
`// by ID`

`G=gap.SmallGroup(180,6);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-5,-3,10,1022,462,723,488,3004]);`
`// Polycyclic`

`G:=Group<a,b,c|a^9=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;`
`// generators/relations`

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