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G = C3×C18order 54 = 2·33

Abelian group of type [3,18]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C18, SmallGroup(54,9)

Series: Derived Chief Lower central Upper central

C1 — C3×C18
C1C3C32C3×C9 — C3×C18
C1 — C3×C18
C1 — C3×C18

Generators and relations for C3×C18
 G = < a,b | a3=b18=1, ab=ba >


Smallest permutation representation of C3×C18
Regular action on 54 points
Generators in S54
(1 41 20)(2 42 21)(3 43 22)(4 44 23)(5 45 24)(6 46 25)(7 47 26)(8 48 27)(9 49 28)(10 50 29)(11 51 30)(12 52 31)(13 53 32)(14 54 33)(15 37 34)(16 38 35)(17 39 36)(18 40 19)
(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)

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

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

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

54 conjugacy classes

class 1  2 3A···3H6A···6H9A···9R18A···18R
order123···36···69···918···18
size111···11···11···11···1

54 irreducible representations

dim11111111
type++
imageC1C2C3C3C6C6C9C18
kernelC3×C18C3×C9C18C3×C6C9C32C6C3
# reps1162621818

Matrix representation of C3×C18 in GL2(𝔽19) generated by

70
011
,
50
03
G:=sub<GL(2,GF(19))| [7,0,0,11],[5,0,0,3] >;

C3×C18 in GAP, Magma, Sage, TeX

C_3\times C_{18}
% in TeX

G:=Group("C3xC18");
// GroupNames label

G:=SmallGroup(54,9);
// by ID

G=gap.SmallGroup(54,9);
# by ID

G:=PCGroup([4,-2,-3,-3,-3,77]);
// Polycyclic

G:=Group<a,b|a^3=b^18=1,a*b=b*a>;
// generators/relations

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