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C72×D165 #30633575

order16170
isoclinismisoclinic class of D165
automorphism?
homologyAbelianization (H1) = C2×C72 · Schur multiplier (H2) = C7

Aliases

direct productC72×D165
unambiguous semidirect productC3⋊(C72×D55)
unambiguous semidirect productC5⋊(C72×D33)
unambiguous semidirect productC11⋊(C72×D15)

Splittings

Fitting, socle, HallC3×C5×C72×C11 ⋊ C2
derived, HallC3×C5×C11 ⋊ C2×C72
HallC5×C72×C11 ⋊ D3
HallC3×C72×C11 ⋊ D5
HallC3×C5×C72 ⋊ D11
HallC72×C11 ⋊ D15
HallC5×C72 ⋊ D33
HallC3×C72 ⋊ D55
HallC5×C11 ⋊ D3×C72
HallC3×C11 ⋊ C72×D5
HallC3×C5 ⋊ C72×D11
SylowC11 ⋊ C72×D15
SylowC5 ⋊ C72×D33
SylowC3 ⋊ C72×D55

Subgroups

subgroupG/H
center, hypercenterC72D165
commutator, cosocle, hypocenterC3×C5×C11C2×C72
FrattiniC1G
Fitting, socleC3×C5×C72×C11C2
LCSG ▷ C3×C5×C11
UCSC1 ◁ C72

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 8085

Conjugacy classes & character table

4116 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC72×D11×D15, C7×D7×D165, C7×D77⋊D15, …
maximal normal subgroup ofC72×D11×D15, C7×D7×D165, C7×D77⋊D15, …
maximal quotient ofC72×Dic165, C2×C72×D165

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 120);