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Cayley diagram

C4×C3⋊S4 #64260

order288
isoclinismisoclinic class of C3⋊S4
automorphismAut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C3⋊S4 · Out(G) = C22×D3
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3

Aliases

direct productC4×C3⋊S4

Splittings

FittingC22×C3×C4 ⋊ D3
derivedC3×A4 ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC4C3⋊S4
commutator, hypocenterC3×A4C2×C4
FrattiniC2C2×C3⋊S4
FittingC22×C3×C4D3
socleC23×C3C2×D3
prime radicalC2×C3⋊S4C2
cosocleC2×C3×A4C22
LCSG ▷ C3×A4
UCSC1 ◁ C4

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 11 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C3⋊S4)⋊C8, #10593176, C8×C3⋊S4, …
maximal normal subgroup of(C3⋊S4)⋊C8, #10593176, C8×C3⋊S4, …
maximal quotient ofC4×C3⋊#226, #10593622, C4×C3⋊GL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 908);