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ΓL(1,𝔽8)×C7⋊AGL(1,𝔽11) #30633508

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

Aliases

direct productΓL(1,𝔽8)×C7⋊AGL(1,𝔽11)
unambiguous semidirect product(C11⋊C5)⋊(D7×ΓL(1,𝔽8))
unambiguous semidirect product(ΓL(1,𝔽8)×D77)⋊C5

Splittings

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

Subgroups

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

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 462

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×C7⋊AGL(1,𝔽11), (ΓL(1,𝔽8)⋊D7)⋊AGL(1,𝔽11), D7×ΓL(1,𝔽8)×AGL(1,𝔽11), …
maximal normal subgroup ofAGL(1,𝔽7)×C7⋊AGL(1,𝔽11), (ΓL(1,𝔽8)⋊D7)⋊AGL(1,𝔽11), D7×ΓL(1,𝔽8)×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, 53);