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G = C5×C10order 50 = 2·52

Abelian group of type [5,10]

direct product, abelian, monomial, 5-elementary

Aliases: C5×C10, SmallGroup(50,5)

Series: Derived Chief Lower central Upper central

C1 — C5×C10
C1C5C52 — C5×C10
C1 — C5×C10
C1 — C5×C10

Generators and relations for C5×C10
 G = < a,b | a5=b10=1, ab=ba >


Smallest permutation representation of C5×C10
Regular action on 50 points
Generators in S50
(1 21 35 43 19)(2 22 36 44 20)(3 23 37 45 11)(4 24 38 46 12)(5 25 39 47 13)(6 26 40 48 14)(7 27 31 49 15)(8 28 32 50 16)(9 29 33 41 17)(10 30 34 42 18)
(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)

G:=sub<Sym(50)| (1,21,35,43,19)(2,22,36,44,20)(3,23,37,45,11)(4,24,38,46,12)(5,25,39,47,13)(6,26,40,48,14)(7,27,31,49,15)(8,28,32,50,16)(9,29,33,41,17)(10,30,34,42,18), (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)>;

G:=Group( (1,21,35,43,19)(2,22,36,44,20)(3,23,37,45,11)(4,24,38,46,12)(5,25,39,47,13)(6,26,40,48,14)(7,27,31,49,15)(8,28,32,50,16)(9,29,33,41,17)(10,30,34,42,18), (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) );

G=PermutationGroup([(1,21,35,43,19),(2,22,36,44,20),(3,23,37,45,11),(4,24,38,46,12),(5,25,39,47,13),(6,26,40,48,14),(7,27,31,49,15),(8,28,32,50,16),(9,29,33,41,17),(10,30,34,42,18)], [(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)])

50 conjugacy classes

class 1  2 5A···5X10A···10X
order125···510···10
size111···11···1

50 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC5×C10C52C10C5
# reps112424

Matrix representation of C5×C10 in GL2(𝔽11) generated by

30
01
,
20
02
G:=sub<GL(2,GF(11))| [3,0,0,1],[2,0,0,2] >;

C5×C10 in GAP, Magma, Sage, TeX

C_5\times C_{10}
% in TeX

G:=Group("C5xC10");
// GroupNames label

G:=SmallGroup(50,5);
// by ID

G=gap.SmallGroup(50,5);
# by ID

G:=PCGroup([3,-2,-5,-5]);
// Polycyclic

G:=Group<a,b|a^5=b^10=1,a*b=b*a>;
// generators/relations

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