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Cayley diagram

C3×Dic27 #66318

order324
isoclinismisoclinic class of D27
automorphismAut(G) = C22×AGL(1,ℤ/27) (order 1944) · Inn(G) = D27 · Out(G) = C22×C9
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productC3×Dic27

Splittings

derivedC27 ⋊ C3×C4
SylowC3×C27 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C3D27
commutator, hypocenterC27C3×C4
FrattiniC2×C9C3×D3
FittingC2×C3×C27C2
socle, prime radicalC2×C32D9
cosocleC2×C27C2×C3
LCSG ▷ C27
UCSC1 ◁ C2×C3

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 34
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 2 generators on 34 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)(31,34,33,32);(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

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10619539, (C3⋊D27)⋊C4, D3×Dic27, …
maximal normal subgroup of#10619539, (C3⋊D27)⋊C4, D3×Dic27, …
maximal quotient ofC3×C27⋊C8, C2×C3×Dic27, C9×Dic27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 10);