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Cayley diagram

A4×Dic6 #64270

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

Aliases

direct productA4×Dic6
unambiguous semidirect productC3⋊(Q8×A4)
unambiguous semidirect product(C22×Dic6)⋊C3

Subgroups

subgroupG/H
center, FrattiniC2C2×D3×A4
commutator, socle, cosocleC23×C3C22×C3
FittingC22×C3×C4C2×C3
prime radicalC2×C3×A4C22
hypercenterC4D3×A4
hypocenterC22×C3C3×Q8
LCSG ▷ C23×C3 ▷ C22×C3
UCSC1 ◁ C2 ◁ C4

98 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:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593187, #10593188, #10593189, …
maximal normal subgroup of#10593187, #10593188, #10593189, …
maximal quotient ofSL(2,𝔽3)×Dic6, #5569⋊C3, #5570⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 918);