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AGL(1,ℤ/8)×C3⋊(C3⋊D3) #13179897

order1728
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25×C33

Aliases

direct productAGL(1,ℤ/8)×C3⋊(C3⋊D3)

Splittings

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

Subgroups

subgroupG/H
centerC2C2×D4×C3⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC33×AGL(1,ℤ/8)C2
socleC2×C33C22×D4
hypercenterAGL(1,ℤ/8)C3⋊(C3⋊D3)
hypocenterC33C2×AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

4172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 216

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×C3⋊D3×AGL(1,ℤ/8)
maximal normal subgroup ofD3×C3⋊D3×AGL(1,ℤ/8)
maximal quotient ofC3⋊(C3⋊D3)×#442

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 45960);