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Cayley diagram

C2×D3×3−1+2 #66449

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) = C22×C32 · Schur multiplier (H2) = C2

Aliases

direct productC2×D3×3−1+2
unambiguous semidirect productC3⋊(C22×3−1+2)
unambiguous semidirect product(C2×C3)⋊(C2×3−1+2)
unambiguous semidirect product(C2×D3×C9)⋊C3

Splittings

FittingC2×C3×3−1+2 ⋊ C2
SylowC3×3−1+2 ⋊ C22

Subgroups

subgroupG/H
centerC2×C3C32×D3
commutator, cosocleC32C22×C32
FrattiniC3C2×C32×D3
FittingC2×C3×3−1+2C2
socleC2×C32C2×C32
prime radicalC2×C32×D3C3
hypercenterC2×3−1+2D3
hypocenterC3C22×3−1+2
LCSG ▷ C32 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 14 points:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×3−1+2⋊C4, #6194⋊C3, #6195⋊C3, …
maximal normal subgroup ofD3×3−1+2⋊C4, #6194⋊C3, #6195⋊C3, …
maximal quotient ofDic6×3−1+2, C4×D3×3−1+2, D12×3−1+2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 141);