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G = C3×D19order 114 = 2·3·19

Direct product of C3 and D19

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C19 — C3×D19
C1C19C57 — C3×D19
C19 — C3×D19
C1C3

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

19C2
19C6

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

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

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

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

C3×D19 is a maximal subgroup of   C57.C6

33 conjugacy classes

class 1  2 3A3B6A6B19A···19I57A···57R
order12336619···1957···57
size1191119192···22···2

33 irreducible representations

dim111122
type+++
imageC1C2C3C6D19C3×D19
kernelC3×D19C57D19C19C3C1
# reps1122918

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

100
010
,
35
2428
,
2828
139
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

Export

Subgroup lattice of C3×D19 in TeX

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