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Cayley diagram

C9×S4 #6252

order216
isoclinismisoclinic class of S4
automorphismAut(G) = C2×C3×S4 (order 144) · Inn(G) = S4 · Out(G) = C2×C3
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C2

Aliases

direct productC9×S4
unambiguous semidirect productA4⋊(C2×C9)

Splittings

FittingC22×C9 ⋊ D3
derivedA4 ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC9S4
commutator, hypocenterA4C2×C9
FrattiniC3C3×S4
FittingC22×C9D3
socleC22×C3C3×D3
prime radicalC3×S4C3
cosocleC3×A4C2×C3
LCSG ▷ A4
UCSC1 ◁ C9

37 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

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

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD9×S4, C2×C9×S4, S4×C27, …
maximal normal subgroup ofD9×S4, C2×C9×S4, S4×C27, …
maximal quotient ofC9×#226, C9×GL(2,𝔽3), C9×SL(2,ℤ/4), …
automorphism group ofC22⋊C27, Q8⋊C27, C2×C22⋊C27
outer automorphism group ofC22×D19, C22×D27

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 89);