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Cayley diagram

C3×C3⋊S4 #6327

order216
isoclinismisoclinic class of C3⋊S4
automorphismAut(G) = C2×((C3⋊S4)⋊C3)⋊C2 (order 864) · Inn(G) = C3⋊S4 · Out(G) = C2×D3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C3⋊S4
unambiguous semidirect productC3⋊(C3×S4)
unambiguous semidirect productA4⋊(C3×D3)

Splittings

Fitting, socleC22×C32 ⋊ D3
derivedC3×A4 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC3C3⋊S4
commutator, cosocle, hypocenterC3×A4C2×C3
FrattiniC1G
Fitting, socleC22×C32D3
LCSG ▷ C3×A4
UCSC1 ◁ C3

77 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 10 points:

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

Conjugacy classes & character table

27 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D3×S4, D3×C3⋊S4, #89101, …
maximal normal subgroup ofC3×D3×S4, D3×C3⋊S4, #89101, …
maximal quotient ofC3×C3⋊#226, C3×C3⋊GL(2,𝔽3), C3×C3⋊SL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 164);