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Cayley diagram

C3⋊GU(2,𝔽3) #63756

order288
isoclinismisoclinic class of C3⋊#226
automorphismAut(G) = C22×((C3⋊S4)⋊C3)⋊C2 (order 1728) · Inn(G) = C3⋊S4 · Out(G) = C22×D3
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊GU(2,𝔽3)

Splittings

derivedC3×SL(2,𝔽3) ⋊ C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC4C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C4
FittingC3×#41D3
socleC2×C3SL(2,ℤ/4)
prime radical, cosocleC3×#41⋊C3C2
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C4

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 27 points:

(1,3,6,10)(2,8)(4,12,13,21)(5,16)(7,24,9,17)(11,14,23,15)(18,22,19,20)(26,27);(25,26,27);(1,4,12)(2,7,11)(3,9,17)(5,15,20)(6,14,23)(8,19,21)(10,18,22)(13,24,16)

Polycyclic presentation:

2^2,3^2,2^3|a^2=b,b^2=g,e^2=g,f^2=g,c^a=c^2,d^a=d^2,e^a=f,f^a=e,e^c=fg,f^c=ef,f^e=fg

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10593219, #10593221, D3×GU(2,𝔽3), …
maximal normal subgroup of#10593219, #10593221, D3×GU(2,𝔽3), …
maximal quotient ofC3⋊(SL(2,𝔽3)⋊C8), C2×C3⋊GU(2,𝔽3), C9⋊GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 404);