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Cayley diagram

C3×C81 #6876

order243
nilpotency class1
isoclinismisoclinic class of C1
automorphismAut(G) = C2×#92319 (order 972) · Inn(G) = C1 · Out(G) = C2×#92319
homologyAbelianization (H1) = G · Schur multiplier (H2) = C3

Aliases

abelianC3×C81

Subgroups

subgroupG/H
centerGC1
commutatorC1G
Frattini, cosocleC27C32
socle, prime radicalC32C27
LCSG ▷ C1
UCSC1 ◁ G
LECSG ▷ C27 ▷ C9 ▷ C3 ▷ C1

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 84
transitive degreeμt(G) = 243

A minimal faithful permutation representation — 2 generators on 84 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)

Conjugacy classes & character table

243 conjugacy classes — view the full character table.

Relationships

center ofC3×D3×C81, #14546454, C32⋊C243, …
derived subgroup ofC3⋊D81, C3⋊Dic81, D3×D81, …
Frattini subgroup ofC9×C243, #14546454, #14546455, …
Fitting subgroup ofC3×D81, D3×C81, C3⋊D81, …
maximal subgroup ofC3×D81, D3×C81, C3⋊D81, …
maximal normal subgroup ofC3×D81, D3×C81, C3⋊D81, …
maximal quotient ofC2×C3×C81, C9×C81, C32⋊C81, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(243, 23);