direct product, abelian, monomial, 5-elementary
Aliases: C5xC10, SmallGroup(50,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C5xC10 |
| C1 — C5xC10 |
| C1 — C5xC10 |
Generators and relations for C5xC10
G = < a,b | a5=b10=1, ab=ba >
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 | ··· | 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 | C5xC10 | C52 | C10 | C5 |
| # reps | 1 | 1 | 24 | 24 |
Matrix representation of C5xC10 ►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] >;
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
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