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Cayley diagram

D3×3−1+2 #4201

order162
isoclinismisoclinic class of D3×H(3,𝔽3)
automorphismAut(G) = D3×#272 (order 324) · Inn(G) = C32×D3 · Out(G) = D3
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C1

Aliases

direct productD3×3−1+2
unambiguous semidirect productC3⋊(C2×3−1+2)
unambiguous semidirect product(D3×C9)⋊C3

Splittings

Fitting, SylowC3×3−1+2 ⋊ C2

Subgroups

subgroupG/H
center, FrattiniC3C32×D3
commutator, socle, cosocleC32C2×C32
FittingC3×3−1+2C2
prime radicalC32×D3C3
hypercenter3−1+2D3
hypocenterC3C2×3−1+2
LCSG ▷ C32 ▷ C3
UCSC1 ◁ C3 ◁ 3−1+2

43 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 18

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,ℤ/9), C2×D3×3−1+2, D9×3−1+2, …
maximal normal subgroup ofD3×AGL(1,ℤ/9), C2×D3×3−1+2, D3×AΣL(1,𝔽27), …
maximal quotient ofDic3×3−1+2, C2×D3×3−1+2, D3×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 37);