Finite Group Databasebeta

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#137×(C3⋊D3)⋊C3 #13153515

order1728
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C3

Aliases

direct product#137×(C3⋊D3)⋊C3
unambiguous semidirect product(C3⋊D3×#137)⋊C3

Splittings

FittingH(3,𝔽3)×#137 ⋊ C2
SylowH(3,𝔽3) ⋊ C2×#137

Subgroups

subgroupG/H
centerC2C2×D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingH(3,𝔽3)×#137C2
socleC2×C3C2×C3×D3×D4
prime radical#41×(C3⋊D3)⋊C3C2
hypercenter#137(C3⋊D3)⋊C3
hypocenterC32C2×C3×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19578);