p-group, cyclic, elementary abelian, simple, monomial
Aliases: C61, also denoted Z61, SmallGroup(61,1)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
| C1 — C61 |
| C1 — C61 |
| C1 — C61 |
| C1 — C61 |
Generators and relations for C61
G = < a | a61=1 >
Smallest permutation representation of C61
►Regular action on 61 points
Generators in S61
(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)
G:=sub<Sym(61)| (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)>;
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) );
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)]])
C61 is a maximal subgroup of
D61 C61⋊C3 C61⋊C5
61 conjugacy classes
| class | 1 | 61A | ··· | 61BH |
| order | 1 | 61 | ··· | 61 |
| size | 1 | 1 | ··· | 1 |
61 irreducible representations
| dim | 1 | 1 |
| type | + | |
| image | C1 | C61 |
| kernel | C61 | C1 |
| # reps | 1 | 60 |
Matrix representation of C61 ►in GL1(𝔽367) generated by
| 137 |
G:=sub<GL(1,GF(367))| [137] >;
C61 in GAP, Magma, Sage, TeX
C_{61} % in TeX
G:=Group("C61"); // GroupNames label
G:=SmallGroup(61,1);
// by ID
G=gap.SmallGroup(61,1);
# by ID
G:=PCGroup([1,-61]:ExponentLimit:=1);
// Polycyclic
G:=Group<a|a^61=1>;
// generators/relations
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