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C3×C13⋊C3×ΓL(1,𝔽32) #31827449

order18135
automorphism?
homologyAbelianization (H1) = C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC3×C13⋊C3×ΓL(1,𝔽32)
unambiguous semidirect productC13⋊(C32×ΓL(1,𝔽32))
unambiguous semidirect product(C3×C13)⋊(C3×ΓL(1,𝔽32))
unambiguous semidirect product(C3×C31×C13⋊C3)⋊C5

Splittings

Fitting, socleC3×C13×C31 ⋊ C3×C5
derived, HallC13×C31 ⋊ C32×C5
HallC3×C31×C13⋊C3 ⋊ C5
HallC13×ΓL(1,𝔽32) ⋊ C32
SylowC31 ⋊ C3×C5×C13⋊C3
SylowC13 ⋊ C32×ΓL(1,𝔽32)

Subgroups

subgroupG/H
center, hypercenterC3C13⋊C3×ΓL(1,𝔽32)
commutator, cosocle, hypocenterC13×C31C32×C5
FrattiniC1G
Fitting, socleC3×C13×C31C3×C5
LCSG ▷ C13×C31
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1209

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×(C13⋊C3)⋊(D31⋊C5), C3×D13⋊C3×ΓL(1,𝔽32), C3×C13⋊C3×D31⋊C5, …
maximal normal subgroup ofC3×(C13⋊C3)⋊(D31⋊C5), C3×D13⋊C3×ΓL(1,𝔽32), C3×C13⋊C3×D31⋊C5, …
maximal quotient ofC2×C3×C13⋊C3×ΓL(1,𝔽32)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18135, 16);