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## G = C3×D19order 114 = 2·3·19

### Direct product of C3 and D19

Aliases: C3×D19, C193C6, C572C2, SmallGroup(114,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C3×D19
 Chief series C1 — C19 — C57 — C3×D19
 Lower central C19 — C3×D19
 Upper central C1 — C3

Generators and relations for C3×D19
G = < a,b,c | a3=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C3×D19
On 57 points
Generators in S57
(1 45 33)(2 46 34)(3 47 35)(4 48 36)(5 49 37)(6 50 38)(7 51 20)(8 52 21)(9 53 22)(10 54 23)(11 55 24)(12 56 25)(13 57 26)(14 39 27)(15 40 28)(16 41 29)(17 42 30)(18 43 31)(19 44 32)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 26)(21 25)(22 24)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(39 50)(40 49)(41 48)(42 47)(43 46)(44 45)(51 57)(52 56)(53 55)

G:=sub<Sym(57)| (1,45,33)(2,46,34)(3,47,35)(4,48,36)(5,49,37)(6,50,38)(7,51,20)(8,52,21)(9,53,22)(10,54,23)(11,55,24)(12,56,25)(13,57,26)(14,39,27)(15,40,28)(16,41,29)(17,42,30)(18,43,31)(19,44,32), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(51,57)(52,56)(53,55)>;

G:=Group( (1,45,33)(2,46,34)(3,47,35)(4,48,36)(5,49,37)(6,50,38)(7,51,20)(8,52,21)(9,53,22)(10,54,23)(11,55,24)(12,56,25)(13,57,26)(14,39,27)(15,40,28)(16,41,29)(17,42,30)(18,43,31)(19,44,32), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)(51,57)(52,56)(53,55) );

G=PermutationGroup([[(1,45,33),(2,46,34),(3,47,35),(4,48,36),(5,49,37),(6,50,38),(7,51,20),(8,52,21),(9,53,22),(10,54,23),(11,55,24),(12,56,25),(13,57,26),(14,39,27),(15,40,28),(16,41,29),(17,42,30),(18,43,31),(19,44,32)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,26),(21,25),(22,24),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(39,50),(40,49),(41,48),(42,47),(43,46),(44,45),(51,57),(52,56),(53,55)]])

C3×D19 is a maximal subgroup of   C57.C6

33 conjugacy classes

 class 1 2 3A 3B 6A 6B 19A ··· 19I 57A ··· 57R order 1 2 3 3 6 6 19 ··· 19 57 ··· 57 size 1 19 1 1 19 19 2 ··· 2 2 ··· 2

33 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C3 C6 D19 C3×D19 kernel C3×D19 C57 D19 C19 C3 C1 # reps 1 1 2 2 9 18

Matrix representation of C3×D19 in GL2(𝔽37) generated by

 10 0 0 10
,
 3 5 24 28
,
 28 28 13 9
G:=sub<GL(2,GF(37))| [10,0,0,10],[3,24,5,28],[28,13,28,9] >;

C3×D19 in GAP, Magma, Sage, TeX

C_3\times D_{19}
% in TeX

G:=Group("C3xD19");
// GroupNames label

G:=SmallGroup(114,4);
// by ID

G=gap.SmallGroup(114,4);
# by ID

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

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

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