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C32×C22⋊SL(2,𝔽3) #11724641

order864
automorphism?
homologyAbelianization (H1) = C33 · Schur multiplier (H2) = C23×C33

Aliases

direct productC32×C22⋊SL(2,𝔽3)

Splittings

FittingC22×C32×Q8 ⋊ C3
derived, SylowC22×Q8 ⋊ C33

Subgroups

subgroupG/H
center, hypercenterC2×C32C22⋊A4
commutator, cosocle, hypocenterC22×Q8C33
FrattiniC2C32×C22⋊A4
FittingC22×C32×Q8C3
socleC23×C32A4
LCSG ▷ C22×Q8
UCSC1 ◁ C2×C32

390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 216

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊(C3⋊#5930), C3⋊(C3⋊#5931)
maximal subgroup ofC3×C3⋊(#395⋊C3), C3×C3⋊(#393⋊C3), C3⋊(C3⋊(#395⋊C3)), …
maximal normal subgroup ofC3×C3⋊(#395⋊C3), C3×C3⋊(#393⋊C3), C3⋊(C3⋊(#395⋊C3)), …
maximal quotient ofC32×#543⋊C3, C32×Q8⋊SL(2,𝔽3), C2×C32×C22⋊SL(2,𝔽3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 4608);