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Cayley diagram

C3×D27 #4171

order162
isoclinismisoclinic class of D27
automorphismAut(G) = C2×AGL(1,ℤ/27) (order 972) · Inn(G) = D27 · Out(G) = C2×C9
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D27

Splittings

Fitting, SylowC3×C27 ⋊ C2
derivedC27 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC3D27
commutator, cosocle, hypocenterC27C2×C3
FrattiniC9C3×D3
FittingC3×C27C2
socleC32D9
LCSG ▷ C27
UCSC1 ◁ C3

19 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 30 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D27, C2×C3×D27, C9×D27, …
maximal normal subgroup ofD3×D27, C2×C3×D27, C9×D27, …
maximal quotient ofC3×Dic27, C2×C3×D27, C9×D27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 7);