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Cayley diagram

C32×D27 #92266

order486
isoclinismisoclinic class of D27
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC32×D27

Splittings

Fitting, SylowC32×C27 ⋊ C2
derivedC27 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC32D27
commutator, cosocle, hypocenterC27C2×C32
FrattiniC9C32×D3
FittingC32×C27C2
socleC33D9
LCSG ▷ C27
UCSC1 ◁ C32

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 2 generators on 33 points:

(4,5,6)(8,11)(9,15)(10,18)(12,13)(14,24)(16,17)(19,26)(20,25)(21,31)(22,30)(23,29)(27,28)(32,33);(1,2,3)(7,26,20,27,31,29,9,8,30,10,33,24,16,13,12,17,14,32,18,22,11,15,23,21,28,25,19)

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×D27, C3⋊D3×D27, C2×C32×D27, …
maximal normal subgroup ofC3×D3×D27, C3⋊D3×D27, C2×C32×D27, …
maximal quotient ofC32×Dic27, C2×C32×D27, C3×C9×D27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(486, 111);