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C9⋊(C22⋊(C22⋊A4)) #13180114

order1728
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C29

Aliases

unambiguous semidirect productC9⋊(C22⋊(C22⋊A4))

Splittings

FittingC26×C9 ⋊ C3
SylowC26 ⋊ 3−1+2

Subgroups

subgroupG/H
center, FrattiniC3C3×C22⋊(C22⋊A4)
commutator, socle, cosocleC26×C3C32
FittingC26×C9C3
prime radicalC3×C22⋊(C22⋊A4)C3
hypercenterC9C22⋊(C22⋊A4)
hypocenterC263−1+2
LCSG ▷ C26×C3 ▷ C26
UCSC1 ◁ C3 ◁ C9

3133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 216

Polycyclic presentation:

3^3,2^6|b^3=c,b^a=bc,d^a=de,e^a=d,f^a=fg,g^a=f,h^a=hi,i^a=h

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 46177);