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ΓL(1,𝔽8)⋊(C7⋊AGL(1,𝔽11)) #30633504

order16170
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊(C7⋊AGL(1,𝔽11))
unambiguous semidirect product(C11⋊C5)⋊(ΓL(1,𝔽8)⋊D7)
unambiguous semidirect product(C11⋊(ΓL(1,𝔽8)⋊D7))⋊C5

Splittings

Fitting, socle, derived, HallC72×C11 ⋊ C2×C3×C5
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C2
HallC7⋊(C7⋊AGL(1,𝔽11)) ⋊ C3
HallC11⋊(ΓL(1,𝔽8)⋊D7) ⋊ C5
HallC72×C11⋊C5 ⋊ C2×C3
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC7⋊D77 ⋊ C3×C5
HallC7×ΓL(1,𝔽8) ⋊ AGL(1,𝔽11)
HallC11⋊C5 ⋊ ΓL(1,𝔽8)⋊D7
SylowC72 ⋊ C3×AGL(1,𝔽11)
SylowC11 ⋊ C5×ΓL(1,𝔽8)⋊D7

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC72×C11C2×C3×C5
LCSG ▷ C72×C11
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 231

Polycyclic presentation:

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

Conjugacy classes & character table

145 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽11)×ΓL(1,𝔽8)⋊D7, D7×ΓL(1,𝔽8)⋊AGL(1,𝔽11), AGL(1,𝔽7)×C7⋊AGL(1,𝔽11), …
maximal normal subgroup ofAGL(1,𝔽11)×ΓL(1,𝔽8)⋊D7, D7×ΓL(1,𝔽8)⋊AGL(1,𝔽11), AGL(1,𝔽7)×C7⋊AGL(1,𝔽11), …
maximal quotient ofΓL(1,𝔽8)⋊(C7⋊(Dic11⋊C5)), C2×ΓL(1,𝔽8)⋊(C7⋊AGL(1,𝔽11))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 49);