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Cayley diagram

D27⋊C3 #4173

order162
isoclinismisoclinic class of D27⋊C3
automorphismAut(G) = AGL(1,ℤ/27) (order 486) · Inn(G) = G · Out(G) = C3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productD27⋊C3
unambiguous semidirect product(C27⋊C3)⋊C2

Splittings

Fitting, SylowC27⋊C3 ⋊ C2
derivedC27 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC27C2×C3
FrattiniC9C3×D3
FittingC27⋊C3C2
socleC3C3×D9
LCSG ▷ C27
UCSC1

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

2,3^4|c^3=d^2e,d^3=e^2,c^a=c^2d,d^a=d^2e,e^a=e^2,c^b=ce

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D27⋊C3, AGL(1,ℤ/27), #6869⋊C2, …
maximal normal subgroup ofC2×D27⋊C3, AGL(1,ℤ/27), C3×D27⋊C3, …
maximal quotient of(C27⋊C3)⋊C4, C2×D27⋊C3, #6864⋊C2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 9);