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Cayley diagram

C3×((C3⋊D3)⋊C3)⋊C2 #66425

order324
isoclinismisoclinic class of ((C3⋊D3)⋊C3)⋊C2
automorphismAut(G) = C2×#1088⋊C2 (order 432) · Inn(G) = ((C3⋊D3)⋊C3)⋊C2 · Out(G) = C22
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×((C3⋊D3)⋊C3)⋊C2
unambiguous semidirect product((C3⋊D3)⋊C3)⋊(C2×C3)

Splittings

Fitting, SylowC3×H(3,𝔽3) ⋊ C22
derivedH(3,𝔽3) ⋊ C22×C3

Subgroups

subgroupG/H
center, hypercenterC3((C3⋊D3)⋊C3)⋊C2
commutator, cosocle, hypocenterH(3,𝔽3)C22×C3
FrattiniC3C3×D32
FittingC3×H(3,𝔽3)C22
socleC32D32
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C3

103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 18

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,3)(5,11,9,7,8,12)(6,10);(4,5,7)(6,8,11,10,9,12)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#1088⋊C2, (((C3⋊D3)⋊C3)⋊C2)⋊D3, D3×((C3⋊D3)⋊C3)⋊C2, …
maximal normal subgroup ofC3×#1088⋊C2, (((C3⋊D3)⋊C3)⋊C2)⋊D3, D3×((C3⋊D3)⋊C3)⋊C2, …
maximal quotient ofC3×#6196, C3×((C3⋊D3)⋊C3)⋊C4, C3×#6198, …
automorphism group ofAΣL(1,𝔽27), #731, (C3⋊D3)⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 117);