Finite Group Databasebeta

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C22×C3×C5×A4 #10624364

order720
isoclinismisoclinic class of A4
automorphism?
homologyAbelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C22×C3

Aliases

direct productC22×C3×C5×A4

Splittings

Fitting, socleC24×C3×C5 ⋊ C3
derivedC22 ⋊ C22×C32×C5
HallC24×C5 ⋊ C32
SylowC24 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC22×C3×C5A4
commutator, cosocle, hypocenterC22C22×C32×C5
FrattiniC1G
Fitting, socleC24×C3×C5C3
LCSG ▷ C22
UCSC1 ◁ C22×C3×C5

196 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 ofC2×C3×C5×SL(2,ℤ/4), C3×C5×GL(2,ℤ/4), C2×C3×A4⋊Dic5, …
maximal normal subgroup ofC2×C3×C5×SL(2,ℤ/4), C3×C5×GL(2,ℤ/4), C2×C3×A4⋊Dic5, …
maximal quotient ofC2×C3×C4×C5×A4, C3×C5×D4×A4, C3×C5×Q8×A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 833);