direct product, cyclic, abelian, monomial
Aliases: C65, also denoted Z65, SmallGroup(65,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C65 |
| C1 — C65 |
| C1 — C65 |
Generators and relations for C65
G = < a | a65=1 >
Smallest permutation representation of C65
►Regular action on 65 points
Generators in S65
(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 53 54 55 56 57 58 59 60 61 62 63 64 65)
G:=sub<Sym(65)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65)]])
C65 is a maximal subgroup of
D65
65 conjugacy classes
| class | 1 | 5A | 5B | 5C | 5D | 13A | ··· | 13L | 65A | ··· | 65AV |
| order | 1 | 5 | 5 | 5 | 5 | 13 | ··· | 13 | 65 | ··· | 65 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C13 | C65 |
| kernel | C65 | C13 | C5 | C1 |
| # reps | 1 | 4 | 12 | 48 |
Matrix representation of C65 ►in GL1(𝔽131) generated by
| 15 |
G:=sub<GL(1,GF(131))| [15] >;
C65 in GAP, Magma, Sage, TeX
C_{65} % in TeX
G:=Group("C65"); // GroupNames label
G:=SmallGroup(65,1);
// by ID
G=gap.SmallGroup(65,1);
# by ID
G:=PCGroup([2,-5,-13]);
// Polycyclic
G:=Group<a|a^65=1>;
// generators/relations
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