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Cayley diagram

C3×A4⋊C8 #63750

order288
isoclinismisoclinic class of S4
automorphismAut(G) = C23×S4 (order 192) · Inn(G) = S4 · Out(G) = C23
homologyAbelianization (H1) = C3×C8 · Schur multiplier (H2) = C2

Aliases

direct productC3×A4⋊C8
unambiguous semidirect productA4⋊(C3×C8)

Splittings

derivedA4 ⋊ C3×C8

Subgroups

subgroupG/H
center, hypercenterC3×C4S4
commutator, hypocenterA4C3×C8
FrattiniC4C3×S4
FittingC22×C3×C4D3
socleC23×C3Dic3
prime radicalC2×C3×A4C4
cosocleC4×A4C2×C3
LCSG ▷ A4
UCSC1 ◁ C3×C4

58 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 15 points:

(1,2,4,3)(5,9,11,6,8,12,7,10)(13,14,15);(2,3,4)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C3⋊S4)⋊C8, #10593176, D3×A4⋊C8, …
maximal normal subgroup of(C3⋊S4)⋊C8, #10593176, D3×A4⋊C8, …
maximal quotient ofC3×SL(2,𝔽3)⋊C8, C3×A4⋊C16, C3×#4628, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 398);