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G = C2×C20order 40 = 23·5

Abelian group of type [2,20]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C20, SmallGroup(40,9)

Series: Derived Chief Lower central Upper central

C1 — C2×C20
C1C2C10C20 — C2×C20
C1 — C2×C20
C1 — C2×C20

Generators and relations for C2×C20
 G = < a,b | a2=b20=1, ab=ba >


Smallest permutation representation of C2×C20
Regular action on 40 points
Generators in S40
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)
(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)

G:=sub<Sym(40)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35), (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)>;

G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35), (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) );

G=PermutationGroup([(1,36),(2,37),(3,38),(4,39),(5,40),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35)], [(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)])

40 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D10A···10L20A···20P
order12224444555510···1020···20
size1111111111111···11···1

40 irreducible representations

dim11111111
type+++
imageC1C2C2C4C5C10C10C20
kernelC2×C20C20C2×C10C10C2×C4C4C22C2
# reps121448416

Matrix representation of C2×C20 in GL2(𝔽41) generated by

10
040
,
210
05
G:=sub<GL(2,GF(41))| [1,0,0,40],[21,0,0,5] >;

C2×C20 in GAP, Magma, Sage, TeX

C_2\times C_{20}
% in TeX

G:=Group("C2xC20");
// GroupNames label

G:=SmallGroup(40,9);
// by ID

G=gap.SmallGroup(40,9);
# by ID

G:=PCGroup([4,-2,-2,-5,-2,80]);
// Polycyclic

G:=Group<a,b|a^2=b^20=1,a*b=b*a>;
// generators/relations

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