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G = C64order 64 = 26

Cyclic group

p-group, cyclic, abelian, monomial

Aliases: C64, also denoted Z64, SmallGroup(64,1)

Series: Derived Chief Lower central Upper central Jennings

C1 — C64
C1C2C4C8C16C32 — C64
C1 — C64
C1 — C64
C1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2C4C4C4C4C4C4C4C4C8C8C8C8C16C16C32 — C64

Generators and relations for C64
 G = < a | a64=1 >


Smallest permutation representation of C64
Regular action on 64 points
Generators in S64
(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 62 63 64)

G:=sub<Sym(64)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)])

C64 is a maximal subgroup of
C128  D64  SD128  Q128
 C2p.C32: M7(2)  C3⋊C64  C52C64  C5⋊C64  C7⋊C64 ...
C64 is a maximal quotient of
C128  C3⋊C64  C52C64  C5⋊C64  C7⋊C64

64 conjugacy classes

class 1  2 4A4B8A8B8C8D16A···16H32A···32P64A···64AF
order1244888816···1632···3264···64
size111111111···11···11···1

64 irreducible representations

dim1111111
type++
imageC1C2C4C8C16C32C64
kernelC64C32C16C8C4C2C1
# reps112481632

Matrix representation of C64 in GL1(𝔽193) generated by

154
G:=sub<GL(1,GF(193))| [154] >;

C64 in GAP, Magma, Sage, TeX

C_{64}
% in TeX

G:=Group("C64");
// GroupNames label

G:=SmallGroup(64,1);
// by ID

G=gap.SmallGroup(64,1);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-2,12,31,50,69,88]);
// Polycyclic

G:=Group<a|a^64=1>;
// generators/relations

Export

Subgroup lattice of C64 in TeX

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