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C3⋊(C5⋊(C7⋊D77)) #30633571

order16170
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C7

Aliases

unambiguous semidirect productC3⋊(C5⋊(C7⋊D77))
unambiguous semidirect productC5⋊(C3⋊(C7⋊D77))
unambiguous semidirect productC7⋊D1155
unambiguous semidirect productC11⋊(C3⋊(C5⋊(C7⋊D7)))

Splittings

Fitting, socle, derived, HallC3×C5×C72×C11 ⋊ C2
HallC5×C72×C11 ⋊ D3
HallC3×C72×C11 ⋊ D5
HallC3×C5×C72 ⋊ D11
HallC72×C11 ⋊ D15
HallC5×C72 ⋊ D33
HallC3×C5×C11 ⋊ C7⋊D7
HallC3×C72 ⋊ D55
HallC5×C11 ⋊ C3⋊(C7⋊D7)
SylowC72 ⋊ D165
HallC3×C11 ⋊ C5⋊(C7⋊D7)
HallC3×C5 ⋊ C7⋊D77
SylowC11 ⋊ C3⋊(C5⋊(C7⋊D7))
SylowC5 ⋊ C3⋊(C7⋊D77)
SylowC3 ⋊ C5⋊(C7⋊D77)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C72×C11C2
LCSG ▷ C3×C5×C72×C11
UCSC1

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 8085

Polycyclic presentation:

2,3,5,7^2,11|b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^6,f^a=f^10

Conjugacy classes & character table

4044 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD7×D1155, D77×D105, D11×C3⋊(C5⋊(C7⋊D7)), …
maximal normal subgroup ofD7×D1155, D77×D105, D11×C3⋊(C5⋊(C7⋊D7)), …
maximal quotient ofC3⋊(C7⋊Dic385), C2×C3⋊(C5⋊(C7⋊D77))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 116);