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## G = C19⋊C12order 228 = 22·3·19

### The semidirect product of C19 and C12 acting via C12/C2=C6

Aliases: C19⋊C12, C38.C6, Dic19⋊C3, C19⋊C3⋊C4, C2.(C19⋊C6), (C2×C19⋊C3).C2, SmallGroup(228,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C19⋊C12
 Chief series C1 — C19 — C38 — C2×C19⋊C3 — C19⋊C12
 Lower central C19 — C19⋊C12
 Upper central C1 — C2

Generators and relations for C19⋊C12
G = < a,b | a19=b12=1, bab-1=a8 >

Character table of C19⋊C12

 class 1 2 3A 3B 4A 4B 6A 6B 12A 12B 12C 12D 19A 19B 19C 38A 38B 38C size 1 1 19 19 19 19 19 19 19 19 19 19 6 6 6 6 6 6 ρ1 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 linear of order 2 ρ3 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ4 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ5 1 1 ζ32 ζ3 -1 -1 ζ32 ζ3 ζ65 ζ6 ζ6 ζ65 1 1 1 1 1 1 linear of order 6 ρ6 1 1 ζ3 ζ32 -1 -1 ζ3 ζ32 ζ6 ζ65 ζ65 ζ6 1 1 1 1 1 1 linear of order 6 ρ7 1 -1 1 1 -i i -1 -1 -i i -i i 1 1 1 -1 -1 -1 linear of order 4 ρ8 1 -1 1 1 i -i -1 -1 i -i i -i 1 1 1 -1 -1 -1 linear of order 4 ρ9 1 -1 ζ32 ζ3 -i i ζ6 ζ65 ζ43ζ3 ζ4ζ32 ζ43ζ32 ζ4ζ3 1 1 1 -1 -1 -1 linear of order 12 ρ10 1 -1 ζ3 ζ32 -i i ζ65 ζ6 ζ43ζ32 ζ4ζ3 ζ43ζ3 ζ4ζ32 1 1 1 -1 -1 -1 linear of order 12 ρ11 1 -1 ζ3 ζ32 i -i ζ65 ζ6 ζ4ζ32 ζ43ζ3 ζ4ζ3 ζ43ζ32 1 1 1 -1 -1 -1 linear of order 12 ρ12 1 -1 ζ32 ζ3 i -i ζ6 ζ65 ζ4ζ3 ζ43ζ32 ζ4ζ32 ζ43ζ3 1 1 1 -1 -1 -1 linear of order 12 ρ13 6 6 0 0 0 0 0 0 0 0 0 0 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 orthogonal lifted from C19⋊C6 ρ14 6 6 0 0 0 0 0 0 0 0 0 0 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 orthogonal lifted from C19⋊C6 ρ15 6 6 0 0 0 0 0 0 0 0 0 0 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 orthogonal lifted from C19⋊C6 ρ16 6 -6 0 0 0 0 0 0 0 0 0 0 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 symplectic faithful, Schur index 2 ρ17 6 -6 0 0 0 0 0 0 0 0 0 0 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 symplectic faithful, Schur index 2 ρ18 6 -6 0 0 0 0 0 0 0 0 0 0 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 -ζ1918-ζ1912-ζ1911-ζ198-ζ197-ζ19 -ζ1917-ζ1916-ζ1914-ζ195-ζ193-ζ192 -ζ1915-ζ1913-ζ1910-ζ199-ζ196-ζ194 symplectic faithful, Schur index 2

Smallest permutation representation of C19⋊C12
On 76 points
Generators in S76
```(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 46 47 48 49 50 51 52 53 54 55 56 57)(58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76)
(1 58 20 39)(2 70 31 57 8 66 21 51 12 76 27 47)(3 63 23 56 15 74 22 44 4 75 34 55)(5 68 26 54 10 71 24 49 7 73 29 52)(6 61 37 53 17 60 25 42 18 72 36 41)(9 59 32 50 19 65 28 40 13 69 38 46)(11 64 35 48 14 62 30 45 16 67 33 43)```

`G:=sub<Sym(76)| (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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,58,20,39)(2,70,31,57,8,66,21,51,12,76,27,47)(3,63,23,56,15,74,22,44,4,75,34,55)(5,68,26,54,10,71,24,49,7,73,29,52)(6,61,37,53,17,60,25,42,18,72,36,41)(9,59,32,50,19,65,28,40,13,69,38,46)(11,64,35,48,14,62,30,45,16,67,33,43)>;`

`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)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57)(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76), (1,58,20,39)(2,70,31,57,8,66,21,51,12,76,27,47)(3,63,23,56,15,74,22,44,4,75,34,55)(5,68,26,54,10,71,24,49,7,73,29,52)(6,61,37,53,17,60,25,42,18,72,36,41)(9,59,32,50,19,65,28,40,13,69,38,46)(11,64,35,48,14,62,30,45,16,67,33,43) );`

`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),(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57),(58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76)], [(1,58,20,39),(2,70,31,57,8,66,21,51,12,76,27,47),(3,63,23,56,15,74,22,44,4,75,34,55),(5,68,26,54,10,71,24,49,7,73,29,52),(6,61,37,53,17,60,25,42,18,72,36,41),(9,59,32,50,19,65,28,40,13,69,38,46),(11,64,35,48,14,62,30,45,16,67,33,43)])`

C19⋊C12 is a maximal subgroup of   Dic38⋊C3  C4×C19⋊C6  D38⋊C6
C19⋊C12 is a maximal quotient of   C19⋊C24

Matrix representation of C19⋊C12 in GL6(𝔽229)

 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 228 108 17 8 17 108
,
 101 158 170 181 114 79 14 109 200 101 1 13 207 219 135 191 176 197 150 115 48 59 71 137 117 83 185 92 227 73 56 56 124 137 79 56

`G:=sub<GL(6,GF(229))| [0,0,0,0,0,228,1,0,0,0,0,108,0,1,0,0,0,17,0,0,1,0,0,8,0,0,0,1,0,17,0,0,0,0,1,108],[101,14,207,150,117,56,158,109,219,115,83,56,170,200,135,48,185,124,181,101,191,59,92,137,114,1,176,71,227,79,79,13,197,137,73,56] >;`

C19⋊C12 in GAP, Magma, Sage, TeX

`C_{19}\rtimes C_{12}`
`% in TeX`

`G:=Group("C19:C12");`
`// GroupNames label`

`G:=SmallGroup(228,1);`
`// by ID`

`G=gap.SmallGroup(228,1);`
`# by ID`

`G:=PCGroup([4,-2,-3,-2,-19,24,3459,679]);`
`// Polycyclic`

`G:=Group<a,b|a^19=b^12=1,b*a*b^-1=a^8>;`
`// generators/relations`

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