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AGL(1,𝔽7)×AGL(1,𝔽11) #17987641

order4620
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productAGL(1,𝔽7)×AGL(1,𝔽11)
unambiguous semidirect productD7⋊(C3×AGL(1,𝔽11))
unambiguous semidirect productD11⋊(C5×AGL(1,𝔽7))
unambiguous semidirect product(D11×AGL(1,𝔽7))⋊C5
unambiguous semidirect product(D7×AGL(1,𝔽11))⋊C3
unambiguous semidirect product(ΓL(1,𝔽8)×AGL(1,𝔽11))⋊C2
unambiguous semidirect product(AGL(1,𝔽7)×C11⋊C5)⋊C2
unambiguous semidirect product(ΓL(1,𝔽8)⋊AGL(1,𝔽11))⋊C2

Splittings

Fitting, socle, derived, HallC7×C11 ⋊ C22×C3×C5
HallD7×AGL(1,𝔽11) ⋊ C3
HallΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallD11×AGL(1,𝔽7) ⋊ C5
HallC7×C11⋊C5 ⋊ C22×C3
HallD7×D11 ⋊ C3×C5
HallC11×ΓL(1,𝔽8) ⋊ C22×C5
HallC11⋊C5 ⋊ C2×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×AGL(1,𝔽7)
SylowC7 ⋊ C2×C3×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C11C22×C3×C5
LCSG ▷ C7×C11
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 18
transitive degreeμt(G) = 77

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×AGL(1,𝔽11), AGL(1,𝔽11)×C3⋊AGL(1,𝔽7), AGL(1,𝔽7)×C3⋊AGL(1,𝔽11), …
maximal normal subgroup ofC3×AGL(1,𝔽7)×AGL(1,𝔽11), C5×AGL(1,𝔽7)×AGL(1,𝔽11), C7×AGL(1,𝔽7)×AGL(1,𝔽11), …
maximal quotient ofAGL(1,𝔽11)×D7⋊C9, C3×AGL(1,𝔽7)×AGL(1,𝔽11), AGL(1,𝔽11)×C3⋊AGL(1,𝔽7), …
automorphism group ofD77
holomorph ofC7×C11

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4620, 38);