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G = C5xC10order 50 = 2·52

Abelian group of type [5,10]

direct product, abelian, monomial, 5-elementary

Aliases: C5xC10, SmallGroup(50,5)

Series: Derived Chief Lower central Upper central

C1 — C5xC10
C1C5C52 — C5xC10
C1 — C5xC10
C1 — C5xC10

Generators and relations for C5xC10
 G = < a,b | a5=b10=1, ab=ba >

Subgroups: 16, all normal (4 characteristic)
Quotients: C1, C2, C5, C10, C52, C5xC10

Smallest permutation representation of C5xC10
Regular action on 50 points
Generators in S50
(1 21 20 43 33)(2 22 11 44 34)(3 23 12 45 35)(4 24 13 46 36)(5 25 14 47 37)(6 26 15 48 38)(7 27 16 49 39)(8 28 17 50 40)(9 29 18 41 31)(10 30 19 42 32)
(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)| (1,21,20,43,33)(2,22,11,44,34)(3,23,12,45,35)(4,24,13,46,36)(5,25,14,47,37)(6,26,15,48,38)(7,27,16,49,39)(8,28,17,50,40)(9,29,18,41,31)(10,30,19,42,32), (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( (1,21,20,43,33)(2,22,11,44,34)(3,23,12,45,35)(4,24,13,46,36)(5,25,14,47,37)(6,26,15,48,38)(7,27,16,49,39)(8,28,17,50,40)(9,29,18,41,31)(10,30,19,42,32), (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([[(1,21,20,43,33),(2,22,11,44,34),(3,23,12,45,35),(4,24,13,46,36),(5,25,14,47,37),(6,26,15,48,38),(7,27,16,49,39),(8,28,17,50,40),(9,29,18,41,31),(10,30,19,42,32)], [(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)]])

C5xC10 is a maximal subgroup of   C52:6C4

50 conjugacy classes

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

50 irreducible representations

dim1111
type++
imageC1C2C5C10
kernelC5xC10C52C10C5
# reps112424

Matrix representation of C5xC10 in GL2(F11) generated by

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

C5xC10 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

Export

Subgroup lattice of C5xC10 in TeX

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