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G = C2×C18order 36 = 22·32

Abelian group of type [2,18]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C18, SmallGroup(36,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C18
C1C3C9C18 — C2×C18
C1 — C2×C18
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 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 19)(17 20)(18 21)
(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)| (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,19)(17,20)(18,21), (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( (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,19)(17,20)(18,21), (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([(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,19),(17,20),(18,21)], [(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 2A2B2C3A3B6A···6F9A···9F18A···18R
order1222336···69···918···18
size1111111···11···11···1

36 irreducible representations

dim111111
type++
imageC1C2C3C6C9C18
kernelC2×C18C18C2×C6C6C22C2
# reps1326618

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

10
018
,
150
011
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

Export

Subgroup lattice of C2×C18 in TeX

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