direct product, cyclic, abelian, monomial
Aliases: C60, also denoted Z60, SmallGroup(60,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C60 |
| C1 — C60 |
| C1 — C60 |
Generators and relations for C60
G = < a | a60=1 >
Smallest permutation representation of C60
►Regular action on 60 points
Generators in S60
(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)
G:=sub<Sym(60)| (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)>;
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) );
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)]])
C60 is a maximal subgroup of
C15⋊3C8 Dic30 D60
60 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | 6B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | ··· | 20H | 30A | ··· | 30H | 60A | ··· | 60P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 |
| kernel | C60 | C30 | C20 | C15 | C12 | C10 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C60 ►in GL1(𝔽61) generated by
| 2 |
G:=sub<GL(1,GF(61))| [2] >;
C60 in GAP, Magma, Sage, TeX
C_{60} % in TeX
G:=Group("C60"); // GroupNames label
G:=SmallGroup(60,4);
// by ID
G=gap.SmallGroup(60,4);
# by ID
G:=PCGroup([4,-2,-3,-5,-2,120]);
// Polycyclic
G:=Group<a|a^60=1>;
// generators/relations
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