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Cayley diagram

C3×C5×A4 #4386

order180
isoclinismisoclinic class of A4
automorphismAut(G) = C4×D3×S4 (order 576) · Inn(G) = A4 · Out(G) = C2×C4×D3
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C5×A4
unambiguous semidirect productC22⋊(C32×C5)
unambiguous semidirect product(C22×C3)⋊(C3×C5)
unambiguous semidirect product(C22×C5)⋊C32
unambiguous semidirect product(C22×C3×C5)⋊C3

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC3×C5A4
commutator, cosocle, hypocenterC22C32×C5
FrattiniC1G
Fitting, socleC22×C3×C5C3
LCSG ▷ C22
UCSC1 ◁ C3×C5

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC3⋊(C5⋊S4), C3⋊(A4⋊Dic5), C3⋊(A4⋊AGL(1,𝔽5)), …
maximal subgroup ofC3×C5×S4, C3×C5⋊S4, C5×C3⋊S4, …
maximal normal subgroup ofC3×C5×S4, C3×C5⋊S4, C5×C3⋊S4, …
maximal quotient ofC3×C5×SL(2,𝔽3), C2×C3×C5×A4, C5×C9×A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(180, 31);