Finite Group Databasebeta

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D51⋊(D31⋊C5) #44539296

order31620
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD51⋊(D31⋊C5)
unambiguous semidirect product(C3⋊(D31⋊C5))⋊D17
unambiguous semidirect product(C17⋊(D31⋊C5))⋊D3
unambiguous semidirect product(D51⋊D31)⋊C5

Splittings

Fitting, socle, derived, HallC3×C17×C31 ⋊ C22×C5
HallC3×C17×ΓL(1,𝔽32) ⋊ C22
HallD51⋊D31 ⋊ C5
HallC17×ΓL(1,𝔽32) ⋊ C2×D3
HallC17×C31 ⋊ C2×C5×D3
HallC3×ΓL(1,𝔽32) ⋊ C2×D17
HallΓL(1,𝔽32) ⋊ D3×D17
HallC3×C31 ⋊ C2×C5×D17
HallC3×C17 ⋊ C2×D31⋊C5
SylowC31 ⋊ C5×D3×D17
SylowC17 ⋊ D3×D31⋊C5
SylowC3 ⋊ D17×D31⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C17×C31C22×C5
LCSG ▷ C3×C17×C31
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1581

Polycyclic presentation:

2^2,5,3,17,31|e^a=e^16,f^a=f^30,d^b=d^2,e^b=e^16,f^c=f^16

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31620, 53);