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C3×#10596244 #13179524

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

Aliases

direct productC3×#10596244

Splittings

FittingC33×AGL(1,ℤ/8) ⋊ C2
SylowC33 ⋊ #575

Subgroups

subgroupG/H
centerC2×C3C2×D4×C3⋊D3
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC33×AGL(1,ℤ/8)C2
socleC2×C33C22×D4
hypercenterC3×AGL(1,ℤ/8)C3⋊D3
hypocenterC32C3×#575
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

1844 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 45587);