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Cayley diagram

D3×C81 #92188

order486
isoclinismisoclinic class of D3
automorphismAut(G) = C2×D3×C27 (order 324) · Inn(G) = D3 · Out(G) = C2×C27
homologyAbelianization (H1) = C2×C81 · Schur multiplier (H2) = C1

Aliases

direct productD3×C81
unambiguous semidirect productC3⋊(C2×C81)

Splittings

Fitting, SylowC3×C81 ⋊ C2
derivedC3 ⋊ C2×C81

Subgroups

subgroupG/H
center, hypercenterC81D3
commutator, hypocenterC3C2×C81
FrattiniC27C3×D3
FittingC3×C81C2
socleC32C2×C27
prime radicalC3×D3C27
cosocleC3×C27C2×C3
LCSG ▷ C3
UCSC1 ◁ C81

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 84
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 2 generators on 84 points:

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

Conjugacy classes & character table

243 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D81, C2×D3×C81, D9×C81, …
maximal normal subgroup ofD3×D81, C2×D3×C81, D3×C243, …
maximal quotient ofDic3×C81, C2×D3×C81, D9×C81, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 33);