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Cayley diagram

C2×C81 #4166

order162
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×C27 (order 54) · Inn(G) = C1 · Out(G) = C2×C27
holomorphG ⋊ Aut(G) = C2×#72221799
homologyAbelianization (H1) = G · Schur multiplier (H2) = C1

Aliases

cyclicC162
abelianC2×C81
multiplicative(ℤ/163)ˣ, (ℤ/243)ˣ, (ℤ/326)ˣ, (ℤ/486)ˣ

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC27C2×C3
socle, prime radicalC2×C3C27
LCSG ▷ C1
UCSC1 ◁ G

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 83
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 1 generator on 83 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)

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

center ofD4×C81, Q8×C81, Dic3×C81, …
derived subgroup ofDic162, D324, #10619489, …
Frattini subgroup ofDic243, C4×C243, Q8⋊C243, …
Fitting subgroup ofDic81, C2×D81
maximal subgroup ofDic81, C4×C81, C2×D81, …
maximal normal subgroup ofDic81, C4×C81, C2×D81, …
maximal quotient ofC4×C81, C22×C81, C2×C243, …
automorphism group ofC163, C243, C2×C163, …
outer automorphism group ofC163, C243, C2×C163, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 2);