Finite Group Databasebeta

← Prev (#26040504) Next (#26040506) →

C5⋊(C5⋊(C11⋊AGL(1,𝔽7))) #26040505

order11550
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C5

Aliases

unambiguous semidirect productC5⋊(C5⋊(C11⋊AGL(1,𝔽7)))
unambiguous semidirect productC11⋊(C5⋊(C5⋊AGL(1,𝔽7)))
unambiguous semidirect productΓL(1,𝔽8)⋊(C5⋊D55)

Splittings

Fitting, socle, derived, HallC52×C7×C11 ⋊ C2×C3
HallC52×C11×ΓL(1,𝔽8) ⋊ C2
HallC5⋊D385 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ D11
HallC52×C11 ⋊ AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ C5⋊D5
HallC52×C7 ⋊ C3×D11
HallC7×C11 ⋊ C3×C5⋊D5
SylowC52 ⋊ C11⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C5⋊D55
SylowC11 ⋊ C5⋊(C5⋊AGL(1,𝔽7))
SylowC7 ⋊ C3×C5⋊D55

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC52×C7×C11C2×C3
LCSG ▷ C52×C7×C11
UCSC1

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

692 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×C5⋊(C11⋊AGL(1,𝔽7)), D55×C5⋊AGL(1,𝔽7), D11×C5⋊(C5⋊AGL(1,𝔽7)), …
maximal normal subgroup ofD5×C5⋊(C11⋊AGL(1,𝔽7)), D55×C5⋊AGL(1,𝔽7), D11×C5⋊(C5⋊AGL(1,𝔽7)), …
maximal quotient ofC11⋊(ΓL(1,𝔽8)⋊#1039), C2×C5⋊(C5⋊(C11⋊AGL(1,𝔽7))), C5⋊(C5⋊(C11⋊(D7⋊C9))), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11550, 69);