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Cayley diagram

C3⋊D3×SL(2,𝔽3) #88979

order432
isoclinismisoclinic class of C3⋊(C3⋊(#41⋊C3))
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

direct productC3⋊D3×SL(2,𝔽3)
unambiguous semidirect productC3⋊(D3×SL(2,𝔽3))

Splittings

Fitting, derivedC32×Q8 ⋊ C2×C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2A4×C3⋊D3
commutator, Fitting, cosocle, hypocenterC32×Q8C2×C3
socleC2×C32C2×A4
LCSG ▷ C32×Q8
UCSC1 ◁ C2

132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13180174, #13180175, #13180176, …
maximal subgroup of#11724156, SL(2,𝔽3)×#157, C3⋊D3×#226, …
maximal normal subgroup of#11724156, SL(2,𝔽3)×#157, C3⋊D3×#226, …
maximal quotient ofSL(2,𝔽3)×C3⋊Dic3, C2×C3⋊D3×SL(2,𝔽3), SL(2,𝔽3)×C3⋊D9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 626);