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## G = C19⋊C6order 114 = 2·3·19

### The semidirect product of C19 and C6 acting faithfully

Aliases: C19⋊C6, D19⋊C3, C19⋊C3⋊C2, SmallGroup(114,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C19⋊C6
 Chief series C1 — C19 — C19⋊C3 — C19⋊C6
 Lower central C19 — C19⋊C6
 Upper central C1

Generators and relations for C19⋊C6
G = < a,b | a19=b6=1, bab-1=a12 >

Character table of C19⋊C6

 class 1 2 3A 3B 6A 6B 19A 19B 19C size 1 19 19 19 19 19 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 ζ32 ζ3 ζ32 ζ3 1 1 1 linear of order 3 ρ4 1 -1 ζ32 ζ3 ζ6 ζ65 1 1 1 linear of order 6 ρ5 1 1 ζ3 ζ32 ζ3 ζ32 1 1 1 linear of order 3 ρ6 1 -1 ζ3 ζ32 ζ65 ζ6 1 1 1 linear of order 6 ρ7 6 0 0 0 0 0 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 orthogonal faithful ρ8 6 0 0 0 0 0 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 orthogonal faithful ρ9 6 0 0 0 0 0 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 orthogonal faithful

Permutation representations of C19⋊C6
On 19 points: primitive - transitive group 19T4
Generators in S19
```(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19)
(2 9 8 19 12 13)(3 17 15 18 4 6)(5 14 10 16 7 11)```

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

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

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

`G:=TransitiveGroup(19,4);`

C19⋊C6 is a maximal subgroup of   F19  D57⋊C3  D19⋊A4
C19⋊C6 is a maximal quotient of   C19⋊C12  C57.C6  D57⋊C3  D19⋊A4

Matrix representation of C19⋊C6 in GL6(𝔽229)

 228 1 0 0 0 0 228 0 1 0 0 0 228 0 0 1 0 0 228 0 0 0 1 0 228 0 0 0 0 1 126 224 23 206 5 102
,
 206 148 92 29 207 127 0 0 0 0 1 0 29 196 51 75 131 102 132 200 148 98 6 1 0 0 1 0 0 0 6 98 148 200 132 103

`G:=sub<GL(6,GF(229))| [228,228,228,228,228,126,1,0,0,0,0,224,0,1,0,0,0,23,0,0,1,0,0,206,0,0,0,1,0,5,0,0,0,0,1,102],[206,0,29,132,0,6,148,0,196,200,0,98,92,0,51,148,1,148,29,0,75,98,0,200,207,1,131,6,0,132,127,0,102,1,0,103] >;`

C19⋊C6 in GAP, Magma, Sage, TeX

`C_{19}\rtimes C_6`
`% in TeX`

`G:=Group("C19:C6");`
`// GroupNames label`

`G:=SmallGroup(114,1);`
`// by ID`

`G=gap.SmallGroup(114,1);`
`# by ID`

`G:=PCGroup([3,-2,-3,-19,974,194]);`
`// Polycyclic`

`G:=Group<a,b|a^19=b^6=1,b*a*b^-1=a^12>;`
`// generators/relations`

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