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Cayley diagram

D3×#272 #66430

order324
isoclinismisoclinic class of D3×#272
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

direct productD3×#272

Splittings

Fitting, derived, SylowC3×H(3,𝔽3) ⋊ C22

Subgroups

subgroupG/H
center, Frattini, hypercenterC3D3×C3⋊D3
commutator, Fitting, cosocle, hypocenterC3×H(3,𝔽3)C22
socleC32C2×C3⋊D3
LCSG ▷ C3×H(3,𝔽3)
UCSC1 ◁ C3

144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#1088, D3×((C3⋊D3)⋊C3)⋊C2, C2×D3×#272, …
maximal normal subgroup ofD3×#1088, D3×((C3⋊D3)⋊C3)⋊C2, C2×D3×#272, …
maximal quotient ofD3×#1084, Dic3×#272, #10619890, …
automorphism group ofC22×3−1+2, D3×3−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 122);