Finite Group Databasebeta

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#10592843⋊C3 #13143260

order1728
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22×C3

Aliases

unambiguous semidirect product#10592843⋊C3

Splittings

FittingH(3,𝔽3)×#131 ⋊ C2
SylowH(3,𝔽3) ⋊ #443

Subgroups

subgroupG/H
centerC8#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C3×C4×#67
prime radical#64159⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C3×#443
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C8 ◁ C2×C8 ◁ #131

433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 9323);