direct product, cyclic, abelian, monomial
Aliases: C52, also denoted Z52, SmallGroup(52,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C52 |
| C1 — C52 |
| C1 — C52 |
Generators and relations for C52
G = < a | a52=1 >
Smallest permutation representation of C52
►Regular action on 52 points
Generators in S52
(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 51 52)
G:=sub<Sym(52)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)]])
C52 is a maximal subgroup of
C13⋊2C8 Dic26 D52
52 conjugacy classes
| class | 1 | 2 | 4A | 4B | 13A | ··· | 13L | 26A | ··· | 26L | 52A | ··· | 52X |
| order | 1 | 2 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C13 | C26 | C52 |
| kernel | C52 | C26 | C13 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 |
Matrix representation of C52 ►in GL1(𝔽53) generated by
| 14 |
G:=sub<GL(1,GF(53))| [14] >;
C52 in GAP, Magma, Sage, TeX
C_{52} % in TeX
G:=Group("C52"); // GroupNames label
G:=SmallGroup(52,2);
// by ID
G=gap.SmallGroup(52,2);
# by ID
G:=PCGroup([3,-2,-13,-2,78]);
// Polycyclic
G:=Group<a|a^52=1>;
// generators/relations
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