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Cayley diagram

C3×C22⋊H(3,𝔽3) #66443

order324
isoclinismisoclinic class of C9⋊A4
automorphism?
homologyAbelianization (H1) = C33 · Schur multiplier (H2) = C2×C34

Aliases

direct productC3×C22⋊H(3,𝔽3)
unambiguous semidirect product(C22×C3)⋊H(3,𝔽3)

Splittings

FittingC22×C33 ⋊ C3
SylowC22 ⋊ C3×H(3,𝔽3)

Subgroups

subgroupG/H
centerC32C3×A4
commutator, cosocleC22×C3C33
FrattiniC3C32×A4
FittingC22×C33C3
socleC22×C32C32
hypercenterC33A4
hypocenterC22C3×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C32 ◁ C33

140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 16 points:

(1,4)(2,3)(9,12,13)(11,15,16);(2,3,4)(8,9,11)(10,13,15)(12,16,14);(5,6,7)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊#6258, C3⋊#88613, ((C3⋊S4)⋊C3)⋊D3, …
maximal subgroup ofC3×(C3⋊S4)⋊C3, C3×#6258, C3⋊((C3⋊S4)⋊C3), …
maximal normal subgroup ofC3×(C3⋊S4)⋊C3, C3×#6258, C3⋊((C3⋊S4)⋊C3), …
maximal quotient ofC3×Q8⋊H(3,𝔽3), C2×C3×C22⋊H(3,𝔽3), C3×#66357, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 135);