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## G = C2×C19⋊C9order 342 = 2·32·19

### Direct product of C2 and C19⋊C9

Aliases: C2×C19⋊C9, C38⋊C9, C192C18, C19⋊C3.2C6, (C2×C19⋊C3).C3, SmallGroup(342,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C2×C19⋊C9
 Chief series C1 — C19 — C19⋊C3 — C19⋊C9 — C2×C19⋊C9
 Lower central C19 — C2×C19⋊C9
 Upper central C1 — C2

Generators and relations for C2×C19⋊C9
G = < a,b,c | a2=b19=c9=1, ab=ba, ac=ca, cbc-1=b5 >

Character table of C2×C19⋊C9

 class 1 2 3A 3B 6A 6B 9A 9B 9C 9D 9E 9F 18A 18B 18C 18D 18E 18F 19A 19B 38A 38B size 1 1 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 9 9 9 9 ρ1 1 1 1 1 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 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 1 linear of order 3 ρ4 1 -1 1 1 -1 -1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ6 ζ65 ζ65 ζ65 ζ6 ζ6 1 1 -1 -1 linear of order 6 ρ5 1 -1 1 1 -1 -1 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ65 ζ6 ζ6 ζ6 ζ65 ζ65 1 1 -1 -1 linear of order 6 ρ6 1 1 1 1 1 1 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 1 linear of order 3 ρ7 1 -1 ζ32 ζ3 ζ6 ζ65 ζ94 ζ97 ζ92 ζ95 ζ98 ζ9 -ζ98 -ζ9 -ζ94 -ζ97 -ζ92 -ζ95 1 1 -1 -1 linear of order 18 ρ8 1 -1 ζ32 ζ3 ζ6 ζ65 ζ9 ζ94 ζ95 ζ98 ζ92 ζ97 -ζ92 -ζ97 -ζ9 -ζ94 -ζ95 -ζ98 1 1 -1 -1 linear of order 18 ρ9 1 1 ζ3 ζ32 ζ3 ζ32 ζ95 ζ92 ζ97 ζ94 ζ9 ζ98 ζ9 ζ98 ζ95 ζ92 ζ97 ζ94 1 1 1 1 linear of order 9 ρ10 1 -1 ζ3 ζ32 ζ65 ζ6 ζ95 ζ92 ζ97 ζ94 ζ9 ζ98 -ζ9 -ζ98 -ζ95 -ζ92 -ζ97 -ζ94 1 1 -1 -1 linear of order 18 ρ11 1 1 ζ32 ζ3 ζ32 ζ3 ζ94 ζ97 ζ92 ζ95 ζ98 ζ9 ζ98 ζ9 ζ94 ζ97 ζ92 ζ95 1 1 1 1 linear of order 9 ρ12 1 1 ζ3 ζ32 ζ3 ζ32 ζ98 ζ95 ζ94 ζ9 ζ97 ζ92 ζ97 ζ92 ζ98 ζ95 ζ94 ζ9 1 1 1 1 linear of order 9 ρ13 1 -1 ζ32 ζ3 ζ6 ζ65 ζ97 ζ9 ζ98 ζ92 ζ95 ζ94 -ζ95 -ζ94 -ζ97 -ζ9 -ζ98 -ζ92 1 1 -1 -1 linear of order 18 ρ14 1 1 ζ32 ζ3 ζ32 ζ3 ζ9 ζ94 ζ95 ζ98 ζ92 ζ97 ζ92 ζ97 ζ9 ζ94 ζ95 ζ98 1 1 1 1 linear of order 9 ρ15 1 1 ζ3 ζ32 ζ3 ζ32 ζ92 ζ98 ζ9 ζ97 ζ94 ζ95 ζ94 ζ95 ζ92 ζ98 ζ9 ζ97 1 1 1 1 linear of order 9 ρ16 1 -1 ζ3 ζ32 ζ65 ζ6 ζ92 ζ98 ζ9 ζ97 ζ94 ζ95 -ζ94 -ζ95 -ζ92 -ζ98 -ζ9 -ζ97 1 1 -1 -1 linear of order 18 ρ17 1 1 ζ32 ζ3 ζ32 ζ3 ζ97 ζ9 ζ98 ζ92 ζ95 ζ94 ζ95 ζ94 ζ97 ζ9 ζ98 ζ92 1 1 1 1 linear of order 9 ρ18 1 -1 ζ3 ζ32 ζ65 ζ6 ζ98 ζ95 ζ94 ζ9 ζ97 ζ92 -ζ97 -ζ92 -ζ98 -ζ95 -ζ94 -ζ9 1 1 -1 -1 linear of order 18 ρ19 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-19/2 -1-√-19/2 -1-√-19/2 -1+√-19/2 complex lifted from C19⋊C9 ρ20 9 -9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-19/2 -1+√-19/2 1-√-19/2 1+√-19/2 complex faithful ρ21 9 -9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√-19/2 -1-√-19/2 1+√-19/2 1-√-19/2 complex faithful ρ22 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√-19/2 -1+√-19/2 -1+√-19/2 -1-√-19/2 complex lifted from C19⋊C9

Smallest permutation representation of C2×C19⋊C9
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 5 17 8 10 18 12 7 6)(3 9 14 15 19 16 4 13 11)(21 24 36 27 29 37 31 26 25)(22 28 33 34 38 35 23 32 30)

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

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

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

Matrix representation of C2×C19⋊C9 in GL9(𝔽2053)

 2052 0 0 0 0 0 0 0 0 0 2052 0 0 0 0 0 0 0 0 0 2052 0 0 0 0 0 0 0 0 0 2052 0 0 0 0 0 0 0 0 0 2052 0 0 0 0 0 0 0 0 0 2052 0 0 0 0 0 0 0 0 0 2052 0 0 0 0 0 0 0 0 0 2052 0 0 0 0 0 0 0 0 0 2052
,
 858 2 1193 856 861 1196 2051 859 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1194 2049 860 338 1190 1716 3 1192 856 1718 5 1191 519 865 1532 2049 1721 1198 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 856 1720 337 2046 523 1201 334 1715 862 1198 334 1715 865 1531 2046 1718 339 1192 0 0 1 0 0 0 0 0 0

G:=sub<GL(9,GF(2053))| [2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052,0,0,0,0,0,0,0,0,0,2052],[858,1,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,1193,0,0,1,0,0,0,0,0,856,0,0,0,1,0,0,0,0,861,0,0,0,0,1,0,0,0,1196,0,0,0,0,0,1,0,0,2051,0,0,0,0,0,0,1,0,859,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0],[1,0,1194,1718,0,0,856,1198,0,0,0,2049,5,1,0,1720,334,0,0,0,860,1191,0,0,337,1715,1,0,0,338,519,0,0,2046,865,0,0,0,1190,865,0,0,523,1531,0,0,1,1716,1532,0,0,1201,2046,0,0,0,3,2049,0,1,334,1718,0,0,0,1192,1721,0,0,1715,339,0,0,0,856,1198,0,0,862,1192,0] >;

C2×C19⋊C9 in GAP, Magma, Sage, TeX

C_2\times C_{19}\rtimes C_9
% in TeX

G:=Group("C2xC19:C9");
// GroupNames label

G:=SmallGroup(342,8);
// by ID

G=gap.SmallGroup(342,8);
# by ID

G:=PCGroup([4,-2,-3,-3,-19,29,583,347]);
// Polycyclic

G:=Group<a,b,c|a^2=b^19=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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