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Cayley diagram

C67 #592

order67
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C3×C11 (order 66) · Inn(G) = C1 · Out(G) = C2×C3×C11
holomorphG ⋊ Aut(G) = AGL(1,𝔽67)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC67

Subgroups

subgroupG/H
centerGC1
commutator, FrattiniC1G
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C1

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 67
transitive degreeμt(G) = 67

A minimal faithful permutation representation — 1 generator on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67)

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

center ofD3×C67, D5×C67, A4×C67, …
derived subgroup ofD67, C67⋊C3, Dic67, …
Frattini subgroup ofC22×C4489, C2×D4489, C5×C4489, …
Fitting subgroup ofD67, C67⋊C3, D67⋊C3, …
socle ofD67, C67⋊C3, D67⋊C3, …
maximal subgroup ofD67, C2×C67, C67⋊C3, …
maximal normal subgroup ofD67, C2×C67, C67⋊C3, …
maximal quotient ofC2×C67, C3×C67, C5×C67, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(67, 1);