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## G = C3×C18order 54 = 2·33

### Abelian group of type [3,18]

Aliases: C3×C18, SmallGroup(54,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18
 Lower central C1 — C3×C18
 Upper central 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 ··· 3H 6A ··· 6H 9A ··· 9R 18A ··· 18R order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C3 C6 C6 C9 C18 kernel C3×C18 C3×C9 C18 C3×C6 C9 C32 C6 C3 # reps 1 1 6 2 6 2 18 18

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

 7 0 0 11
,
 5 0 0 3
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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