direct product, cyclic, abelian, monomial
Aliases: C57, also denoted Z57, SmallGroup(57,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C57 |
| C1 — C57 |
| C1 — C57 |
Generators and relations for C57
G = < a | a57=1 >
Smallest permutation representation of C57
►Regular action on 57 points
Generators in S57
(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)
G:=sub<Sym(57)| (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)>;
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) );
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)]])
C57 is a maximal subgroup of
D57 C19⋊2C9
57 conjugacy classes
| class | 1 | 3A | 3B | 19A | ··· | 19R | 57A | ··· | 57AJ |
| order | 1 | 3 | 3 | 19 | ··· | 19 | 57 | ··· | 57 |
| size | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | |||
| image | C1 | C3 | C19 | C57 |
| kernel | C57 | C19 | C3 | C1 |
| # reps | 1 | 2 | 18 | 36 |
Matrix representation of C57 ►in GL1(𝔽229) generated by
| 193 |
G:=sub<GL(1,GF(229))| [193] >;
C57 in GAP, Magma, Sage, TeX
C_{57} % in TeX
G:=Group("C57"); // GroupNames label
G:=SmallGroup(57,2);
// by ID
G=gap.SmallGroup(57,2);
# by ID
G:=PCGroup([2,-3,-19]);
// Polycyclic
G:=Group<a|a^57=1>;
// generators/relations
Export