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Cayley diagram

D3×H(3,𝔽3) #4199

order162
isoclinismisoclinic class of D3×H(3,𝔽3)
automorphismAut(G) = D3×AGL(2,𝔽3) (order 2592) · Inn(G) = C32×D3 · Out(G) = GL(2,𝔽3)
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32

Aliases

direct productD3×H(3,𝔽3)
unambiguous semidirect productC3⋊(C2×H(3,𝔽3))
unambiguous semidirect product(C32×D3)⋊C3

Splittings

Fitting, SylowC3×H(3,𝔽3) ⋊ C2

Subgroups

subgroupG/H
center, FrattiniC3C32×D3
commutator, socle, cosocleC32C2×C32
FittingC3×H(3,𝔽3)C2
hypercenterH(3,𝔽3)D3
hypocenterC3C2×H(3,𝔽3)
LCSG ▷ C32 ▷ C3
UCSC1 ◁ C3 ◁ H(3,𝔽3)

61 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)(4,5,7)(6,9,11)(8,12,10)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×(C3⋊D3)⋊C3, D3×#272, C2×D3×H(3,𝔽3), …
maximal normal subgroup ofD3×(C3⋊D3)⋊C3, D3×#272, C2×D3×H(3,𝔽3), …
maximal quotient ofDic3×H(3,𝔽3), C2×D3×H(3,𝔽3), D3×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(162, 35);