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## G = C19⋊C3order 57 = 3·19

### The semidirect product of C19 and C3 acting faithfully

Aliases: C19⋊C3, SmallGroup(57,1)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C19⋊C3
G = < a,b | a19=b3=1, bab-1=a11 >

Character table of C19⋊C3

 class 1 3A 3B 19A 19B 19C 19D 19E 19F size 1 19 19 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ3 1 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ4 3 0 0 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ1914+ζ193+ζ192 ζ199+ζ196+ζ194 complex faithful ρ5 3 0 0 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ198 ζ1917+ζ1916+ζ195 complex faithful ρ6 3 0 0 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1915+ζ1913+ζ1910 ζ1911+ζ197+ζ19 complex faithful ρ7 3 0 0 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ1911+ζ197+ζ19 ζ1914+ζ193+ζ192 complex faithful ρ8 3 0 0 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ1917+ζ1916+ζ195 ζ1915+ζ1913+ζ1910 complex faithful ρ9 3 0 0 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ198 complex faithful

Permutation representations of C19⋊C3
On 19 points: primitive - transitive group 19T3
Generators in S19
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)```

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

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

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

`G:=TransitiveGroup(19,3);`

C19⋊C3 is a maximal subgroup of   C19⋊C6  C19⋊C9  C19⋊A4  C133⋊C3  C1334C3
C19⋊C3 is a maximal quotient of   C192C9  C19⋊A4  C133⋊C3  C1334C3

Matrix representation of C19⋊C3 in GL3(𝔽7) generated by

 5 0 5 6 0 3 2 1 6
,
 1 3 0 0 6 1 0 6 0
`G:=sub<GL(3,GF(7))| [5,6,2,0,0,1,5,3,6],[1,0,0,3,6,6,0,1,0] >;`

C19⋊C3 in GAP, Magma, Sage, TeX

`C_{19}\rtimes C_3`
`% in TeX`

`G:=Group("C19:C3");`
`// GroupNames label`

`G:=SmallGroup(57,1);`
`// by ID`

`G=gap.SmallGroup(57,1);`
`# by ID`

`G:=PCGroup([2,-3,-19,85]);`
`// Polycyclic`

`G:=Group<a,b|a^19=b^3=1,b*a*b^-1=a^11>;`
`// generators/relations`

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