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Cayley diagram

C3×3−1+2⋊C4 #66402

order324
isoclinismisoclinic class of AGL(1,ℤ/9)
automorphismAut(G) = C2×D3×AGL(1,ℤ/9) (order 648) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C2×D3
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×3−1+2⋊C4
unambiguous semidirect productDic9⋊C32
unambiguous semidirect product(C3×Dic9)⋊C3

Splittings

derivedC9 ⋊ C32×C4
SylowC3×3−1+2 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2×C3AGL(1,ℤ/9)
commutator, hypocenterC9C32×C4
FrattiniC2×C3C32×D3
FittingC2×C3×3−1+2C2
socleC2×C32C3×D3
prime radicalC2×C33D3
cosocleC2×C9C2×C32
LCSG ▷ C9
UCSC1 ◁ C2×C3

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 16 points:

(5,9,12,7,8,11)(6,10)(13,16,15,14);(1,2,3)(4,12,7,10,8,9,6,5,11)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6189⋊C3, (C3⋊AGL(1,ℤ/9))⋊C4, D3×3−1+2⋊C4, …
maximal normal subgroup of#6189⋊C3, (C3⋊AGL(1,ℤ/9))⋊C4, D3×3−1+2⋊C4, …
maximal quotient ofC3×3−1+2⋊C8, C2×C3×3−1+2⋊C4, C3×#66316, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 94);