direct product, abelian, monomial, 5-elementary
Aliases: C5×C10, SmallGroup(50,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — 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 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
C52⋊6C4
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
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