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#10592523⋊C3 #13141984

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

Aliases

unambiguous semidirect product#10592523⋊C3

Splittings

FittingH(3,𝔽3)×#104 ⋊ C2
SylowH(3,𝔽3) ⋊ #443

Subgroups

subgroupG/H
centerC4D4×(C3⋊D3)⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#104C2
socleC2×C3C3×#67⋊C4
prime radical#64159⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#104(C3⋊D3)⋊C3
hypocenterC32C3×#443
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

415 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,c^a=cg,f^a=fg,h^a=h^2,i^a=i^2,c^b=cf,f^b=fg,f^c=fg,h^d=hi

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, 8047);