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C32×C5×A4 #10587451

order540
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C33×C5 · Schur multiplier (H2) = C2×C33

Aliases

direct productC32×C5×A4
unambiguous semidirect productC22⋊(C33×C5)
unambiguous semidirect product(C22×C3)⋊(C32×C5)
unambiguous semidirect product(C22×C5)⋊C33
unambiguous semidirect product(C22×C3×C5)⋊C32

Splittings

Fitting, socleC22×C32×C5 ⋊ C3
derived, SylowC22 ⋊ C33×C5
HallC22×C5 ⋊ C33

Subgroups

subgroupG/H
center, hypercenterC32×C5A4
commutator, cosocle, hypocenterC22C33×C5
FrattiniC1G
Fitting, socleC22×C32×C5C3
LCSG ▷ C22
UCSC1 ◁ C32×C5

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 180

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊(C3⋊(C5⋊S4))
maximal subgroup ofC32×C5×S4, C32×C5⋊S4, C3×C5×C3⋊S4, …
maximal normal subgroup ofC32×C5×S4, C32×C5⋊S4, C3×C5×C3⋊S4, …
maximal quotient ofC32×C5×SL(2,𝔽3), C2×C32×C5×A4, C3×C5×C9×A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(540, 111);