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Cayley diagram

A4×D12 #64272

order288
automorphismAut(G) = D3×D4×S4 (order 1152) · Inn(G) = C2×D3×A4 · Out(G) = C23
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C22

Aliases

direct productA4×D12
unambiguous semidirect productC4⋊(D3×A4)
unambiguous semidirect product(C22×D12)⋊C3

Splittings

FittingC22×C3×C4 ⋊ C2×C3

Subgroups

subgroupG/H
center, FrattiniC2C2×D3×A4
commutator, socle, cosocleC23×C3C22×C3
FittingC22×C3×C4C2×C3
hypercenterC4D3×A4
hypocenterC22×C3C3×D4
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2 ◁ C4

138 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:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593183, #10593184, #10593185, …
maximal normal subgroup of#10593183, #10593184, #10593185, …
maximal quotient ofC4⋊(C3⋊(#41⋊C3)), SL(2,𝔽3)×D12, #5576⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 920);