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Cayley diagram

C3×C8×A4 #63989

order288
isoclinismisoclinic class of A4
automorphismAut(G) = C22×D3×S4 (order 576) · Inn(G) = A4 · Out(G) = C23×D3
homologyAbelianization (H1) = C32×C8 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×C8×A4
unambiguous semidirect product(C22×C8)⋊C32
unambiguous semidirect product(C22×C3×C8)⋊C3

Splittings

FittingC22×C3×C8 ⋊ C3
derivedC22 ⋊ C32×C8
SylowC22×C8 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC3×C8A4
commutator, hypocenterC22C32×C8
FrattiniC4C2×C3×A4
FittingC22×C3×C8C3
socleC23×C3C3×C4
prime radicalC2×C3×A4C4
cosocleC22×C4C2×C32
LCSG ▷ C22
UCSC1 ◁ C3×C8

68 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:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11920579, #11920580, #11920581, …
maximal subgroup ofC3×A4⋊C16, C3⋊(A4⋊C16), A4×C3⋊C16, …
maximal normal subgroup ofC3×A4⋊C16, C3⋊(A4⋊C16), A4×C3⋊C16, …
maximal quotient ofC3×C8×SL(2,𝔽3), C3×A4×C16, C3×#504⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 637);