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Cayley diagram

C3×C4⋊S4 #64250

order288
automorphismAut(G) = C2×D4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×C4⋊S4
unambiguous semidirect productC4⋊(C3×S4)

Splittings

FittingC22×C3×C4 ⋊ D3

Subgroups

subgroupG/H
centerC2×C3C2×S4
commutator, cosocleC2×A4C22×C3
FrattiniC2C2×C3×S4
FittingC22×C3×C4D3
socleC23×C3C2×D3
hypercenterC3×C4S4
hypocenterA4C3×D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2×C3 ◁ C3×C4

118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593177, #10593179, #10593183, …
maximal normal subgroup of#10593177, #10593179, #10593183, …
maximal quotient ofC3×C4⋊#226, C3×C4⋊GL(2,𝔽3), C3×#5395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 898);