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Cayley diagram

C71 #603

order71
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C5×C7 (order 70) · Inn(G) = C1 · Out(G) = C2×C5×C7
holomorphG ⋊ Aut(G) = AGL(1,𝔽71)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC71

Subgroups

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

2 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 71
transitive degreeμt(G) = 71

A minimal faithful permutation representation — 1 generator on 71 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71)

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

center ofD3×C71, D5×C71, A4×C71, …
derived subgroup ofD71, Dic71, C2×D71, …
Frattini subgroup ofC3×C5041, C5041⋊C5, C5×C5041, …
Fitting subgroup ofD71, C71⋊C5, C71⋊C7, …
socle ofD71, C71⋊C5, C71⋊C7, …
maximal subgroup ofD71, C2×C71, C3×C71, …
maximal normal subgroup ofD71, C2×C71, C3×C71, …
maximal quotient ofC2×C71, C3×C71, C5×C71, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(71, 1);