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G = C56order 56 = 23·7

Cyclic group

direct product, cyclic, abelian, monomial

Aliases: C56, also denoted Z56, SmallGroup(56,2)

Series: Derived Chief Lower central Upper central

C1 — C56
C1C2C4C28 — 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 4A4B7A···7F8A8B8C8D14A···14F28A···28L56A···56X
order12447···7888814···1428···2856···56
size11111···111111···11···11···1

56 irreducible representations

dim11111111
type++
imageC1C2C4C7C8C14C28C56
kernelC56C28C14C8C7C4C2C1
# reps1126461224

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

Export

Subgroup lattice of C56 in TeX

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