direct product, cyclic, abelian, monomial
Aliases: C55, also denoted Z55, SmallGroup(55,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C55 |
| C1 — C55 |
| C1 — C55 |
Generators and relations for C55
G = < a | a55=1 >
Smallest permutation representation of C55
►Regular action on 55 points
Generators in S55
(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)
G:=sub<Sym(55)| (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)>;
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) );
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)]])
C55 is a maximal subgroup of
D55 C11⋊C25
55 conjugacy classes
| class | 1 | 5A | 5B | 5C | 5D | 11A | ··· | 11J | 55A | ··· | 55AN |
| order | 1 | 5 | 5 | 5 | 5 | 11 | ··· | 11 | 55 | ··· | 55 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C5 | C11 | C55 |
| kernel | C55 | C11 | C5 | C1 |
| # reps | 1 | 4 | 10 | 40 |
Matrix representation of C55 ►in GL1(𝔽331) generated by
| 258 |
G:=sub<GL(1,GF(331))| [258] >;
C55 in GAP, Magma, Sage, TeX
C_{55} % in TeX
G:=Group("C55"); // GroupNames label
G:=SmallGroup(55,2);
// by ID
G=gap.SmallGroup(55,2);
# by ID
G:=PCGroup([2,-5,-11]);
// Polycyclic
G:=Group<a|a^55=1>;
// generators/relations
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