Finite Group Databasebeta

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C5×((C31⋊C3)⋊C5)⋊D7 #45654448

order32550
automorphism?
homologyAbelianization (H1) = C2×C3×C52 · Schur multiplier (H2) = C5

Aliases

direct productC5×((C31⋊C3)⋊C5)⋊D7
unambiguous semidirect product((C31⋊C3)⋊C5)⋊(C5×D7)
unambiguous semidirect product((C31⋊C3)⋊D7)⋊C52
unambiguous semidirect product(C5×(C31⋊C3)⋊D7)⋊C5

Splittings

Fitting, socleC5×C7×C31 ⋊ C2×C3×C5
derived, HallC7×C31 ⋊ C2×C3×C52
HallC5×C7×(C31⋊C3)⋊C5 ⋊ C2
HallC5×C7⋊(D31⋊C5) ⋊ C3
HallC5×C7×ΓL(1,𝔽32) ⋊ C2×C3
HallC5×(C31⋊C3)⋊C5 ⋊ D7
Hall(C31⋊C3)⋊D7 ⋊ C52
HallC5×ΓL(1,𝔽32) ⋊ C3×D7
HallC7×C31⋊C3 ⋊ C2×C52
HallD217 ⋊ C3×C52
HallC31⋊C3 ⋊ C52×D7
SylowC31 ⋊ C3×C52×D7
SylowC7 ⋊ C5×AGL(1,𝔽31)

Subgroups

subgroupG/H
center, hypercenterC5((C31⋊C3)⋊C5)⋊D7
commutator, cosocle, hypocenterC7×C31C2×C3×C52
FrattiniC1G
Fitting, socleC5×C7×C31C2×C3×C5
LCSG ▷ C7×C31
UCSC1 ◁ C5

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1085

Conjugacy classes & character table

410 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32550, 84);