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Cayley diagram

C2×C83 #4229

order166
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C41 (order 82) · Inn(G) = C1 · Out(G) = C2×C41
holomorphG ⋊ Aut(G) = C2×AGL(1,𝔽83)
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC166
abelianC2×C83
multiplicative(ℤ/167)ˣ, (ℤ/334)ˣ

Subgroups

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

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 85
transitive degreeμt(G) = 166

A minimal faithful permutation representation — 1 generator on 85 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85)

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

center ofD4×C83, Q8×C83, Dic3×C83, …
derived subgroup ofDic166, D332, C22⋊D83, …
Frattini subgroup ofC4×C6889, Dic6889, Dic13778, …
Fitting subgroup ofDic83, C2×D83, Dic83⋊C41, …
socle ofDic83, C4×C83, C2×D83, …
maximal subgroup ofDic83, C4×C83, C2×D83, …
maximal normal subgroup ofDic83, C4×C83, C2×D83, …
maximal quotient ofC4×C83, C22×C83, D3×C83, …
automorphism group ofC167, C2×C167
outer automorphism group ofC167, C2×C167

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(166, 2);