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Cayley diagram

C2×C3×C4×A4 #64331

order288
isoclinismisoclinic class of A4
automorphismAut(G) = D3×D4×S4 (order 1152) · Inn(G) = A4 · Out(G) = C2×D3×D4
homologyAbelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22×C3

Aliases

direct productC2×C3×C4×A4
unambiguous semidirect product(C23×C4)⋊C32
unambiguous semidirect product(C23×C3×C4)⋊C3

Splittings

FittingC23×C3×C4 ⋊ C3
derivedC22 ⋊ C2×C32×C4
SylowC23×C4 ⋊ C32

Subgroups

subgroupG/H
center, hypercenterC2×C3×C4A4
commutator, hypocenterC22C2×C32×C4
FrattiniC2C22×C3×A4
FittingC23×C3×C4C3
socleC24×C3C2×C3
prime radicalC22×C3×A4C2
cosocleC23C22×C32
LCSG ▷ C22
UCSC1 ◁ C2×C3×C4

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×A4⋊C8, C3×#5409, C3×C4×SL(2,ℤ/4), …
maximal normal subgroup ofC2×C3×A4⋊C8, C3×#5409, C3×C4×SL(2,ℤ/4), …
maximal quotient ofC3×C42×A4, C3×A4×C22⋊C4, C3×A4×C4⋊C4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 979);