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## G = C19⋊C9order 171 = 32·19

### The semidirect product of C19 and C9 acting faithfully

Aliases: C19⋊C9, C19⋊C3.C3, SmallGroup(171,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C19⋊C9
 Chief series C1 — C19 — C19⋊C3 — C19⋊C9
 Lower central C19 — C19⋊C9
 Upper central C1

Generators and relations for C19⋊C9
G = < a,b | a19=b9=1, bab-1=a5 >

Character table of C19⋊C9

 class 1 3A 3B 9A 9B 9C 9D 9E 9F 19A 19B size 1 19 19 19 19 19 19 19 19 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 linear of order 3 ρ3 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 linear of order 3 ρ4 1 ζ32 ζ3 ζ94 ζ92 ζ98 ζ9 ζ97 ζ95 1 1 linear of order 9 ρ5 1 ζ3 ζ32 ζ92 ζ9 ζ94 ζ95 ζ98 ζ97 1 1 linear of order 9 ρ6 1 ζ32 ζ3 ζ9 ζ95 ζ92 ζ97 ζ94 ζ98 1 1 linear of order 9 ρ7 1 ζ32 ζ3 ζ97 ζ98 ζ95 ζ94 ζ9 ζ92 1 1 linear of order 9 ρ8 1 ζ3 ζ32 ζ95 ζ97 ζ9 ζ98 ζ92 ζ94 1 1 linear of order 9 ρ9 1 ζ3 ζ32 ζ98 ζ94 ζ97 ζ92 ζ95 ζ9 1 1 linear of order 9 ρ10 9 0 0 0 0 0 0 0 0 -1+√-19/2 -1-√-19/2 complex faithful ρ11 9 0 0 0 0 0 0 0 0 -1-√-19/2 -1+√-19/2 complex faithful

Permutation representations of C19⋊C9
On 19 points: primitive - transitive group 19T5
Generators in S19
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)
(2 5 17 8 10 18 12 7 6)(3 9 14 15 19 16 4 13 11)```

`G:=sub<Sym(19)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,5,17,8,10,18,12,7,6)(3,9,14,15,19,16,4,13,11)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19), (2,5,17,8,10,18,12,7,6)(3,9,14,15,19,16,4,13,11) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)], [(2,5,17,8,10,18,12,7,6),(3,9,14,15,19,16,4,13,11)]])`

`G:=TransitiveGroup(19,5);`

C19⋊C9 is a maximal subgroup of   F19

Matrix representation of C19⋊C9 in GL9(𝔽2053)

 2052 1 0 0 0 0 0 0 0 2052 0 1 0 0 0 0 0 0 2052 0 0 1 0 0 0 0 0 2052 0 0 0 1 0 0 0 0 2052 0 0 0 0 1 0 0 0 2052 0 0 0 0 0 1 0 0 2052 0 0 0 0 0 0 1 0 2052 0 0 0 0 0 0 0 1 858 1197 1191 1716 5 335 855 861 1195
,
 0 0 0 0 1 0 0 0 0 859 1197 1191 1716 5 335 855 861 1195 337 852 526 1536 329 521 7 334 857 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 4 333 1713 864 1532 2048 1719 1197 2051 2050 1719 1198 1190 1715 4 335 857 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(9,GF(2053))| [2052,2052,2052,2052,2052,2052,2052,2052,858,1,0,0,0,0,0,0,0,1197,0,1,0,0,0,0,0,0,1191,0,0,1,0,0,0,0,0,1716,0,0,0,1,0,0,0,0,5,0,0,0,0,1,0,0,0,335,0,0,0,0,0,1,0,0,855,0,0,0,0,0,0,1,0,861,0,0,0,0,0,0,0,1,1195],[0,859,337,1,0,4,2050,0,0,0,1197,852,0,0,333,1719,1,0,0,1191,526,0,0,1713,1198,0,0,0,1716,1536,0,0,864,1190,0,0,1,5,329,0,0,1532,1715,0,0,0,335,521,0,1,2048,4,0,0,0,855,7,0,0,1719,335,0,1,0,861,334,0,0,1197,857,0,0,0,1195,857,0,0,2051,2,0,0] >;`

C19⋊C9 in GAP, Magma, Sage, TeX

`C_{19}\rtimes C_9`
`% in TeX`

`G:=Group("C19:C9");`
`// GroupNames label`

`G:=SmallGroup(171,3);`
`// by ID`

`G=gap.SmallGroup(171,3);`
`# by ID`

`G:=PCGroup([3,-3,-3,-19,9,326,194]);`
`// Polycyclic`

`G:=Group<a,b|a^19=b^9=1,b*a*b^-1=a^5>;`
`// generators/relations`

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