Copied to
clipboard

## G = C5×C10order 50 = 2·52

### Abelian group of type [5,10]

Aliases: C5×C10, SmallGroup(50,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10
 Chief series C1 — C5 — C52 — C5×C10
 Lower central C1 — C5×C10
 Upper central 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 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)]])

C5×C10 is a maximal subgroup of   C526C4

50 conjugacy classes

 class 1 2 5A ··· 5X 10A ··· 10X order 1 2 5 ··· 5 10 ··· 10 size 1 1 1 ··· 1 1 ··· 1

50 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C5 C10 kernel C5×C10 C52 C10 C5 # reps 1 1 24 24

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

 3 0 0 1
,
 2 0 0 2
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

Export

׿
×
𝔽