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Cayley diagram

Dic3×3−1+2 #66403

order324
isoclinismisoclinic class of D3×H(3,𝔽3)
automorphismAut(G) = C2×D3×#272 (order 648) · Inn(G) = C32×D3 · Out(G) = C2×D3
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C1

Aliases

direct productDic3×3−1+2
unambiguous semidirect productC3⋊(C4×3−1+2)
unambiguous semidirect product(C9×Dic3)⋊C3

Splittings

SylowC3×3−1+2 ⋊ C4

Subgroups

subgroupG/H
center, FrattiniC2×C3C32×D3
commutatorC32C32×C4
FittingC2×C3×3−1+2C2
socle, cosocleC2×C32C2×C32
prime radicalC2×C33C2×C3
hypercenterC2×3−1+2D3
hypocenterC3C4×3−1+2
LCSG ▷ C32 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

70 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:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6189⋊C3, Dic3×AGL(1,ℤ/9), (C3⋊AGL(1,ℤ/9))⋊C4, …
maximal normal subgroup of#6189⋊C3, Dic3×AGL(1,ℤ/9), (C3⋊AGL(1,ℤ/9))⋊C4, …
maximal quotient ofC3⋊C8×3−1+2, C2×Dic3×3−1+2, Dic3×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 95);