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## G = C2×C18order 36 = 22·32

### Abelian group of type [2,18]

Aliases: C2×C18, SmallGroup(36,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18
 Chief series C1 — C3 — C9 — C18 — C2×C18
 Lower central C1 — C2×C18
 Upper central C1 — C2×C18

Generators and relations for C2×C18
G = < a,b | a2=b18=1, ab=ba >

Smallest permutation representation of C2×C18
Regular action on 36 points
Generators in S36
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)
(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)

G:=sub<Sym(36)| (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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)>;

G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26), (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) );

G=PermutationGroup([[(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(17,25),(18,26)], [(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)]])

C2×C18 is a maximal subgroup of   C9⋊D4  C9.A4  C9⋊A4

36 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A ··· 6F 9A ··· 9F 18A ··· 18R order 1 2 2 2 3 3 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1

36 irreducible representations

 dim 1 1 1 1 1 1 type + + image C1 C2 C3 C6 C9 C18 kernel C2×C18 C18 C2×C6 C6 C22 C2 # reps 1 3 2 6 6 18

Matrix representation of C2×C18 in GL2(𝔽19) generated by

 1 0 0 18
,
 15 0 0 11
G:=sub<GL(2,GF(19))| [1,0,0,18],[15,0,0,11] >;

C2×C18 in GAP, Magma, Sage, TeX

C_2\times C_{18}
% in TeX

G:=Group("C2xC18");
// GroupNames label

G:=SmallGroup(36,5);
// by ID

G=gap.SmallGroup(36,5);
# by ID

G:=PCGroup([4,-2,-2,-3,-3,46]);
// Polycyclic

G:=Group<a,b|a^2=b^18=1,a*b=b*a>;
// generators/relations

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