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## G = C2×C19⋊C3order 114 = 2·3·19

### Direct product of C2 and C19⋊C3

Aliases: C2×C19⋊C3, C38⋊C3, C192C6, SmallGroup(114,2)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C19⋊C3
G = < a,b,c | a2=b19=c3=1, ab=ba, ac=ca, cbc-1=b11 >

Character table of C2×C19⋊C3

 class 1 2 3A 3B 6A 6B 19A 19B 19C 19D 19E 19F 38A 38B 38C 38D 38E 38F size 1 1 19 19 19 19 3 3 3 3 3 3 3 3 3 3 3 3 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 ζ6 ζ65 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ4 1 1 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 -1 ζ32 ζ3 ζ65 ζ6 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 6 ρ6 1 1 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ7 3 3 0 0 0 0 ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ1914+ζ193+ζ192 ζ199+ζ196+ζ194 complex lifted from C19⋊C3 ρ8 3 3 0 0 0 0 ζ1911+ζ197+ζ19 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1915+ζ1913+ζ1910 ζ1911+ζ197+ζ19 complex lifted from C19⋊C3 ρ9 3 -3 0 0 0 0 ζ1917+ζ1916+ζ195 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 -ζ1915-ζ1913-ζ1910 -ζ1914-ζ193-ζ192 -ζ1911-ζ197-ζ19 -ζ199-ζ196-ζ194 -ζ1918-ζ1912-ζ198 -ζ1917-ζ1916-ζ195 complex faithful ρ10 3 -3 0 0 0 0 ζ1915+ζ1913+ζ1910 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 -ζ1911-ζ197-ζ19 -ζ199-ζ196-ζ194 -ζ1914-ζ193-ζ192 -ζ1918-ζ1912-ζ198 -ζ1917-ζ1916-ζ195 -ζ1915-ζ1913-ζ1910 complex faithful ρ11 3 -3 0 0 0 0 ζ1918+ζ1912+ζ198 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 -ζ1917-ζ1916-ζ195 -ζ1911-ζ197-ζ19 -ζ1915-ζ1913-ζ1910 -ζ1914-ζ193-ζ192 -ζ199-ζ196-ζ194 -ζ1918-ζ1912-ζ198 complex faithful ρ12 3 3 0 0 0 0 ζ1918+ζ1912+ζ198 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ198 complex lifted from C19⋊C3 ρ13 3 -3 0 0 0 0 ζ1911+ζ197+ζ19 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 -ζ1914-ζ193-ζ192 -ζ1918-ζ1912-ζ198 -ζ199-ζ196-ζ194 -ζ1917-ζ1916-ζ195 -ζ1915-ζ1913-ζ1910 -ζ1911-ζ197-ζ19 complex faithful ρ14 3 -3 0 0 0 0 ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 -ζ1918-ζ1912-ζ198 -ζ1915-ζ1913-ζ1910 -ζ1917-ζ1916-ζ195 -ζ1911-ζ197-ζ19 -ζ1914-ζ193-ζ192 -ζ199-ζ196-ζ194 complex faithful ρ15 3 3 0 0 0 0 ζ1917+ζ1916+ζ195 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1915+ζ1913+ζ1910 ζ1914+ζ193+ζ192 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ198 ζ1917+ζ1916+ζ195 complex lifted from C19⋊C3 ρ16 3 3 0 0 0 0 ζ1914+ζ193+ζ192 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 ζ1911+ζ197+ζ19 ζ1914+ζ193+ζ192 complex lifted from C19⋊C3 ρ17 3 3 0 0 0 0 ζ1915+ζ1913+ζ1910 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ1911+ζ197+ζ19 ζ199+ζ196+ζ194 ζ1914+ζ193+ζ192 ζ1918+ζ1912+ζ198 ζ1917+ζ1916+ζ195 ζ1915+ζ1913+ζ1910 complex lifted from C19⋊C3 ρ18 3 -3 0 0 0 0 ζ1914+ζ193+ζ192 ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ195 ζ1911+ζ197+ζ19 ζ1918+ζ1912+ζ198 ζ1915+ζ1913+ζ1910 -ζ199-ζ196-ζ194 -ζ1917-ζ1916-ζ195 -ζ1918-ζ1912-ζ198 -ζ1915-ζ1913-ζ1910 -ζ1911-ζ197-ζ19 -ζ1914-ζ193-ζ192 complex faithful

Smallest permutation representation of C2×C19⋊C3
On 38 points
Generators in S38
(1 20)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)
(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)
(2 8 12)(3 15 4)(5 10 7)(6 17 18)(9 19 13)(11 14 16)(21 27 31)(22 34 23)(24 29 26)(25 36 37)(28 38 32)(30 33 35)

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

G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38), (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), (2,8,12)(3,15,4)(5,10,7)(6,17,18)(9,19,13)(11,14,16)(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35) );

G=PermutationGroup([(1,20),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38)], [(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)], [(2,8,12),(3,15,4),(5,10,7),(6,17,18),(9,19,13),(11,14,16),(21,27,31),(22,34,23),(24,29,26),(25,36,37),(28,38,32),(30,33,35)])

C2×C19⋊C3 is a maximal subgroup of   C19⋊C12  C38.A4

Matrix representation of C2×C19⋊C3 in GL3(𝔽7) generated by

 6 0 0 0 6 0 0 0 6
,
 6 3 4 5 6 4 3 3 5
,
 1 1 4 0 2 0 0 2 4
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[6,5,3,3,6,3,4,4,5],[1,0,0,1,2,2,4,0,4] >;

C2×C19⋊C3 in GAP, Magma, Sage, TeX

C_2\times C_{19}\rtimes C_3
% in TeX

G:=Group("C2xC19:C3");
// GroupNames label

G:=SmallGroup(114,2);
// by ID

G=gap.SmallGroup(114,2);
# by ID

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

G:=Group<a,b,c|a^2=b^19=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations

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