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Cayley diagram

C2×C3×C5×A4 #67226

order360
isoclinismisoclinic class of A4
automorphismAut(G) = C4×D3×S4 (order 576) · Inn(G) = A4 · Out(G) = C2×C4×D3
homologyAbelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C5×A4
unambiguous semidirect productC23⋊(C32×C5)
unambiguous semidirect product(C23×C3)⋊(C3×C5)
unambiguous semidirect product(C23×C5)⋊C32
unambiguous semidirect product(C23×C3×C5)⋊C3

Splittings

Fitting, socleC23×C3×C5 ⋊ C3
derivedC22 ⋊ C2×C32×C5
HallC23×C5 ⋊ C32
SylowC23 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5A4
commutator, cosocle, hypocenterC22C2×C32×C5
FrattiniC1G
Fitting, socleC23×C3×C5C3
LCSG ▷ C22
UCSC1 ◁ C2×C3×C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 90

A minimal faithful permutation representation — 2 generators on 14 points:

(1,2)(7,8,9)(10,11,12,13,14);(3,4,5)(6,9)(7,8)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13088539, #13088542, #13088544, …
maximal subgroup ofC3×C5×SL(2,ℤ/4), C3×A4⋊Dic5, C5×C3⋊SL(2,ℤ/4), …
maximal normal subgroup ofC3×C5×SL(2,ℤ/4), C3×A4⋊Dic5, C5×C3⋊SL(2,ℤ/4), …
maximal quotient ofC3×C4×C5×A4, C2×C3×C5×SL(2,𝔽3), C3×C5×#41⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 156);