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C3⋊(C5⋊((C11⋊C5)⋊D11)) #31827671

order18150
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C5⋊((C11⋊C5)⋊D11))
unambiguous semidirect productC5⋊(C3⋊((C11⋊C5)⋊D11))
unambiguous semidirect product(C11⋊C5)⋊D165

Splittings

Fitting, socle, derivedC3×C5×C112 ⋊ C2×C5
HallC3×C5×C11×C11⋊C5 ⋊ C2
HallC5×C11×C11⋊C5 ⋊ D3
HallC3×C112 ⋊ C5×D5
SylowC112 ⋊ C5×D15
SylowC3 ⋊ C5⋊((C11⋊C5)⋊D11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C112C2×C5
LCSG ▷ C3×C5×C112
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 825

Polycyclic presentation:

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

Conjugacy classes & character table

585 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD15×(C11⋊C5)⋊D11, D11×C3⋊(C5⋊AGL(1,𝔽11)), AGL(1,𝔽11)×D165, …
maximal normal subgroup ofD15×(C11⋊C5)⋊D11, D11×C3⋊(C5⋊AGL(1,𝔽11)), AGL(1,𝔽11)×D165, …
maximal quotient ofC3⋊((C11⋊C5)⋊Dic55), C2×C3⋊(C5⋊((C11⋊C5)⋊D11))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18150, 155);