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D85⋊(D37⋊C3) #49179804

order37740
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD85⋊(D37⋊C3)
unambiguous semidirect product(C5⋊(D37⋊C3))⋊D17
unambiguous semidirect product(C17⋊(D37⋊C3))⋊D5
unambiguous semidirect product(D85⋊D37)⋊C3

Splittings

Fitting, socle, derived, HallC5×C17×C37 ⋊ C22×C3
HallD85⋊D37 ⋊ C3
HallC5×C17×C37⋊C3 ⋊ C22
HallC17×C37⋊C3 ⋊ C2×D5
HallC17×C37 ⋊ C2×C3×D5
HallC5×C37⋊C3 ⋊ C2×D17
HallC5×C37 ⋊ C2×C3×D17
HallC37⋊C3 ⋊ D5×D17
HallC5×C17 ⋊ C2×D37⋊C3
SylowC37 ⋊ C3×D5×D17
SylowC17 ⋊ D5×D37⋊C3
SylowC5 ⋊ D17×D37⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C17×C37C22×C3
LCSG ▷ C5×C17×C37
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3145

Polycyclic presentation:

2^2,3,5,17,37|e^a=e^16,f^a=f^36,d^b=d^4,e^b=e^16,f^c=f^26

Conjugacy classes & character table

384 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37740, 59);