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C5⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11)) #26040477

order11550
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11))
unambiguous semidirect productΓL(1,𝔽8)⋊(C5⋊AGL(1,𝔽11))
unambiguous semidirect product(C11⋊C5)⋊(C5⋊AGL(1,𝔽7))
unambiguous semidirect product(C5⋊(C11⋊AGL(1,𝔽7)))⋊C5
unambiguous semidirect product(C5⋊(C7⋊AGL(1,𝔽11)))⋊C3

Splittings

Fitting, socle, derivedC5×C7×C11 ⋊ C2×C3×C5
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C2
HallC5⋊(C7⋊AGL(1,𝔽11)) ⋊ C3
HallC5×C7×C11⋊C5 ⋊ C2×C3
HallC5×C11⋊C5 ⋊ AGL(1,𝔽7)
HallC11×ΓL(1,𝔽8) ⋊ C5×D5
HallC7×C11 ⋊ C3×C5×D5
HallΓL(1,𝔽8) ⋊ C5⋊AGL(1,𝔽11)
SylowC11 ⋊ C5×C5⋊AGL(1,𝔽7)
SylowC7 ⋊ C3×C5⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C11C2×C3×C5
LCSG ▷ C5×C7×C11
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽11)×C5⋊AGL(1,𝔽7), AGL(1,𝔽7)×C5⋊AGL(1,𝔽11), D5×ΓL(1,𝔽8)⋊AGL(1,𝔽11), …
maximal normal subgroup ofAGL(1,𝔽11)×C5⋊AGL(1,𝔽7), AGL(1,𝔽7)×C5⋊AGL(1,𝔽11), D5×ΓL(1,𝔽8)⋊AGL(1,𝔽11), …
maximal quotient ofΓL(1,𝔽8)⋊((C11⋊C5)⋊Dic5), C2×C5⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽11)), C5⋊((C11⋊C5)⋊(D7⋊C9)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11550, 41);