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Cayley diagram

D71 #3655

order142
isoclinismisoclinic class of D71
automorphismAut(G) = AGL(1,𝔽71) (order 4970) · Inn(G) = G · Out(G) = C5×C7
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

dihedralD71
unambiguous semidirect productC71⋊C2

Splittings

Fitting, socle, derived, SylowC71 ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC71C2
LCSG ▷ C71
UCSC1

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2,71|b^a=b^70

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D71, C3×D71, D213, …
maximal normal subgroup ofC2×D71, C3×D71, D71⋊C5, …
maximal quotient ofDic71, C2×D71, C3×D71, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(142, 1);