Finite Group Databasebeta

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#123×(C3⋊D3)⋊C3 #13153043

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

Aliases

direct product#123×(C3⋊D3)⋊C3
unambiguous semidirect product(C3⋊D3×#123)⋊C3

Splittings

FittingH(3,𝔽3)×#123 ⋊ C2
SylowH(3,𝔽3) ⋊ C2×#123

Subgroups

subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#123C2
prime radicalC2×D4×(C3⋊D3)⋊C3C2
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C2×C3×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

1275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19106);