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#127⋊H(3,𝔽3) #11721354

order864
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C32×C4

Aliases

unambiguous semidirect product#127⋊H(3,𝔽3)
unambiguous semidirect product(C3×#127⋊C3)⋊C3

Splittings

FittingC32×#127 ⋊ C3
Sylow#127 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC3C3×#127⋊C3
commutator, cosocleC3×C42C2×C32
Frattini, socleC22×C3C2×C3×A4
FittingC32×#127C3
hypercenterC32#127⋊C3
hypocenterC42C2×H(3,𝔽3)
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C3 ◁ C32

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 36

Polycyclic presentation:

2,3^3,2^4|e^2=h,f^2=g,e^a=eh,f^a=fg,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(C3⋊#4626)⋊C3, #13136790, #10593631⋊C3, …
maximal normal subgroup of(C3⋊#4626)⋊C3, #13136790, #10593631⋊C3, …
maximal quotient of#384⋊H(3,𝔽3), #392⋊H(3,𝔽3), #399⋊H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 1321);