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Cayley diagram

C3⋊(D27⋊C3) #92317

order486
isoclinismisoclinic class of C3⋊(D27⋊C3)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(D27⋊C3)

Splittings

Fitting, SylowC3×C27⋊C3 ⋊ C2
derivedC3×C27 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C27C2×C3
FrattiniC9C3×C3⋊D3
FittingC3×C27⋊C3C2
socleC32C3×D9
LCSG ▷ C3×C27
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 3 generators on 30 points:

(2,3)(5,16,19,8,9,18)(6,12)(7,15)(10,21,20,11,22,29)(13,14)(17,30,28,24,27,23)(25,26);(1,2,3);(4,24,18,25,29,22,6,5,28,7,30,16,13,11,10,14,9,27,15,23,8,12,21,20,26,19,17)

Polycyclic presentation:

2,3^5|c^3=e^2f,e^3=f^2,c^a=c^2e,d^a=d^2,e^a=e^2f,f^a=f^2,c^b=cf

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D27⋊C3, C2×C3⋊(D27⋊C3), #10624453⋊C2, …
maximal normal subgroup ofD3×D27⋊C3, C2×C3⋊(D27⋊C3), #10624465⋊C2, …
maximal quotient ofC3⋊((C27⋊C3)⋊C4), C2×C3⋊(D27⋊C3), C3⋊(#6864⋊C2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 162);