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Cayley diagram

C3×C3⋊GL(2,𝔽3) #88970

order432
isoclinismisoclinic class of C3⋊#226
automorphismAut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C3⋊S4 · Out(G) = C22×D3
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

direct productC3×C3⋊GL(2,𝔽3)
unambiguous semidirect productC3⋊(C3×GL(2,𝔽3))

Splittings

FittingC32×Q8 ⋊ D3
derivedC3×SL(2,𝔽3) ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2×C3C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C2×C3
FrattiniC2C3×C3⋊S4
FittingC32×Q8D3
socleC2×C32S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3

117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 14 points:

(1,2,8,3,4,7,6,5)(9,10,11)(13,14);(12,13,14);(2,5,8)(3,6,7)

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#64198, C3×#64199, C3×#64202, …
maximal normal subgroup ofC3×#64198, C3×#64199, C3×#64202, …
maximal quotient ofC3×C3⋊#859, C2×C3×C3⋊GL(2,𝔽3), C9×C3⋊GL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 617);