Finite Group Databasebeta

← Prev (#10624236) Next (#10624238) →

C3×C4×C5×A4 #10624237

order720
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C32×C4×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C4×C5×A4
unambiguous semidirect product(C22×C4)⋊(C32×C5)
unambiguous semidirect product(C22×C3×C4)⋊(C3×C5)
unambiguous semidirect product(C22×C4×C5)⋊C32
unambiguous semidirect product(C22×C3×C4×C5)⋊C3

Splittings

FittingC22×C3×C4×C5 ⋊ C3
derivedC22 ⋊ C32×C4×C5
HallC22×C4×C5 ⋊ C32
SylowC22×C4 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC3×C4×C5A4
commutator, hypocenterC22C32×C4×C5
FrattiniC2C2×C3×C5×A4
FittingC22×C3×C4×C5C3
socleC23×C3×C5C2×C3
prime radicalC2×C3×C5×A4C2
cosocleC23C2×C32×C5
LCSG ▷ C22
UCSC1 ◁ C3×C4×C5

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 180

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C5×A4⋊C8, C3×A4⋊#168, C5×C3⋊(A4⋊C8), …
maximal normal subgroup ofC3×C5×A4⋊C8, C3×A4⋊#168, C5×C3⋊(A4⋊C8), …
maximal quotient ofC3×C4×C5×SL(2,𝔽3), C3×C5×C8×A4, C3×C5×#131⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 706);