direct product, cyclic, abelian, monomial
Aliases: C50, also denoted Z50, SmallGroup(50,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C50 |
| C1 — C50 |
| C1 — C50 |
Generators and relations for C50
G = < a | a50=1 >
Smallest permutation representation of C50
►Regular action on 50 points
Generators in S50
(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)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)]])
C50 is a maximal subgroup of
Dic25
50 conjugacy classes
| class | 1 | 2 | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 25A | ··· | 25T | 50A | ··· | 50T |
| order | 1 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 25 | ··· | 25 | 50 | ··· | 50 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C5 | C10 | C25 | C50 |
| kernel | C50 | C25 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 4 | 4 | 20 | 20 |
Matrix representation of C50 ►in GL2(𝔽101) generated by
| 14 | 0 |
| 0 | 88 |
G:=sub<GL(2,GF(101))| [14,0,0,88] >;
C50 in GAP, Magma, Sage, TeX
C_{50} % in TeX
G:=Group("C50"); // GroupNames label
G:=SmallGroup(50,2);
// by ID
G=gap.SmallGroup(50,2);
# by ID
G:=PCGroup([3,-2,-5,-5,34]);
// Polycyclic
G:=Group<a|a^50=1>;
// generators/relations
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