direct product, cyclic, abelian, monomial
Aliases: C51, also denoted Z51, SmallGroup(51,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C51 |
| C1 — C51 |
| C1 — C51 |
Generators and relations for C51
G = < a | a51=1 >
Smallest permutation representation of C51
►Regular action on 51 points
Generators in S51
(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)
G:=sub<Sym(51)| (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)>;
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) );
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)]])
C51 is a maximal subgroup of
D51
51 conjugacy classes
| class | 1 | 3A | 3B | 17A | ··· | 17P | 51A | ··· | 51AF |
| order | 1 | 3 | 3 | 17 | ··· | 17 | 51 | ··· | 51 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C17 | C51 |
| kernel | C51 | C17 | C3 | C1 |
| # reps | 1 | 2 | 16 | 32 |
Matrix representation of C51 ►in GL1(𝔽103) generated by
| 38 |
G:=sub<GL(1,GF(103))| [38] >;
C51 in GAP, Magma, Sage, TeX
C_{51} % in TeX
G:=Group("C51"); // GroupNames label
G:=SmallGroup(51,1);
// by ID
G=gap.SmallGroup(51,1);
# by ID
G:=PCGroup([2,-3,-17]);
// Polycyclic
G:=Group<a|a^51=1>;
// generators/relations
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