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Cayley diagram

C3⋊(C3⋊GL(2,𝔽3)) #88973

order432
isoclinismisoclinic class of C3⋊(C3⋊#226)
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C33

Aliases

unambiguous semidirect productC3⋊(C3⋊GL(2,𝔽3))

Splittings

FittingC32×Q8 ⋊ D3
derivedC32×SL(2,𝔽3) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C2
FittingC32×Q8D3
socleC2×C32S4
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C2

208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 14 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11724178, #11724179, #11724182, …
maximal normal subgroup of#11724178, #11724179, #11724182, …
maximal quotient ofC3⋊(C3⋊#859), C2×C3⋊(C3⋊GL(2,𝔽3)), C3⋊(C9⋊GL(2,𝔽3)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 620);