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Cayley diagram

C3×C5×S4 #67208

order360
isoclinismisoclinic class of S4
automorphismAut(G) = C2×C4×S4 (order 192) · Inn(G) = S4 · Out(G) = C2×C4
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC3×C5×S4
unambiguous semidirect productA4⋊(C2×C3×C5)

Splittings

Fitting, socleC22×C3×C5 ⋊ D3
derivedA4 ⋊ C2×C3×C5

Subgroups

subgroupG/H
center, hypercenterC3×C5S4
commutator, cosocle, hypocenterA4C2×C3×C5
FrattiniC1G
Fitting, socleC22×C3×C5D3
LCSG ▷ A4
UCSC1 ◁ C3×C5

48 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:

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

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D5×S4, C5×D3×S4, S4×D15, …
maximal normal subgroup ofC3×D5×S4, C5×D3×S4, S4×D15, …
maximal quotient ofC3×C5×#226, C3×C5×GL(2,𝔽3), C3×C5×SL(2,ℤ/4), …
outer automorphism group ofC22×D31

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 138);