direct product, cyclic, abelian, monomial
Aliases: C56, also denoted Z56, SmallGroup(56,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C56 |
| C1 — C56 |
| C1 — C56 |
Generators and relations for C56
G = < a | a56=1 >
Smallest permutation representation of C56
►Regular action on 56 points
Generators in S56
(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)
G:=sub<Sym(56)| (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)>;
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) );
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)]])
C56 is a maximal subgroup of
C7⋊C16 C8⋊D7 C56⋊C2 D56 Dic28
56 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14F | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C4 | C7 | C8 | C14 | C28 | C56 |
| kernel | C56 | C28 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 6 | 4 | 6 | 12 | 24 |
Matrix representation of C56 ►in GL1(𝔽113) generated by
| 82 |
G:=sub<GL(1,GF(113))| [82] >;
C56 in GAP, Magma, Sage, TeX
C_{56} % in TeX
G:=Group("C56"); // GroupNames label
G:=SmallGroup(56,2);
// by ID
G=gap.SmallGroup(56,2);
# by ID
G:=PCGroup([4,-2,-7,-2,-2,56,34]);
// Polycyclic
G:=Group<a|a^56=1>;
// generators/relations
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