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Cayley diagram

C64 #320

order64
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C16 (order 32) · Inn(G) = C1 · Out(G) = C2×C16
holomorphG ⋊ Aut(G) = #72224662
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC64

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC32C2
socle, prime radicalC2C32
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

7 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 1 generator on 64 points:

(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)

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

center ofM256, #13737, C3⋊C128, …
derived subgroup ofD128, SD256, Q256, …
Frattini subgroup ofC128, C2×C128, M256, …
maximal subgroup ofC128, C2×C64, M128, …
maximal normal subgroup ofC128, C2×C64, M128, …
maximal quotient ofC128, C2×C64, C3⋊C64, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 1);