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G = C2xC18order 36 = 22·32

Abelian group of type [2,18]

direct product, abelian, monomial, 2-elementary

Aliases: C2xC18, SmallGroup(36,5)

Series: Derived Chief Lower central Upper central

C1 — C2xC18
C1C3C9C18 — C2xC18
C1 — C2xC18
C1 — C2xC18

Generators and relations for C2xC18
 G = < a,b | a2=b18=1, ab=ba >

Subgroups: 15, all normal (6 characteristic)
Quotients: C1, C2, C3, C22, C6, C9, C2xC6, C18, C2xC18

Smallest permutation representation of C2xC18
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)]])

C2xC18 is a maximal subgroup of   C9:D4  C9.A4  C9:A4

36 conjugacy classes

class 1 2A2B2C3A3B6A···6F9A···9F18A···18R
order1222336···69···918···18
size1111111···11···11···1

36 irreducible representations

dim111111
type++
imageC1C2C3C6C9C18
kernelC2xC18C18C2xC6C6C22C2
# reps1326618

Matrix representation of C2xC18 in GL2(F19) generated by

10
018
,
150
011
G:=sub<GL(2,GF(19))| [1,0,0,18],[15,0,0,11] >;

C2xC18 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

Export

Subgroup lattice of C2xC18 in TeX

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