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#126⋊H(3,𝔽3) #11721351

order864
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32

Aliases

unambiguous semidirect product#126⋊H(3,𝔽3)
unambiguous semidirect product(C3×#126⋊C3)⋊C3

Splittings

FittingC32×#126 ⋊ C3
Sylow#126 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC3C3×#126⋊C3
commutator, cosocleC3×C42C2×C32
Frattini, socleC22×C3C2×C3×A4
FittingC32×#126C3
hypercenterC32#126⋊C3
hypocenterC42C2×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C3 ◁ C32

114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Polycyclic presentation:

2,3^3,2^4|e^2=h,f^2=g,e^a=egh,f^a=fh,c^b=cd,e^b=ef,f^b=eg,g^b=h,h^b=gh

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#13147226, #13147227, C2×#126⋊H(3,𝔽3), …
maximal normal subgroup of#13147226, #13147227, C2×#126⋊H(3,𝔽3), …
maximal quotient of#383⋊H(3,𝔽3), #396⋊H(3,𝔽3), #401⋊H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 1318);