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Cayley diagram

D3×SL(2,𝔽3) #3785

order144
isoclinismisoclinic class of C3⋊(#41⋊C3)
automorphismAut(G) = C2×D3×S4 (order 288) · Inn(G) = D3×A4 · Out(G) = C22
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productD3×SL(2,𝔽3)
unambiguous semidirect productC3⋊(C2×SL(2,𝔽3))
unambiguous semidirect product(D3×Q8)⋊C3

Splittings

Fitting, derivedC3×Q8 ⋊ C2×C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D3×A4
commutator, Fitting, cosocle, hypocenterC3×Q8C2×C3
socleC2×C3C2×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2

39 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

21 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#226, #64201, #64202, …
maximal normal subgroup ofD3×#226, #64201, #64202, …
maximal quotient ofDic3×SL(2,𝔽3), C2×D3×SL(2,𝔽3), D9⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(144, 128);