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C2×ΓL(1,𝔽8)×AGL(1,𝔽11) #17987644

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

Aliases

direct productC2×ΓL(1,𝔽8)×AGL(1,𝔽11)
unambiguous semidirect productD11⋊(C2×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×D11)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(C11⋊C5)⋊(C22×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C11⋊C5)⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C2×ΓL(1,𝔽8)×D11)⋊C5
unambiguous semidirect product(C2×C7×AGL(1,𝔽11))⋊C3

Splittings

Fitting, socleC2×C7×C11 ⋊ C2×C3×C5
derived, HallC7×C11 ⋊ C22×C3×C5
HallC2×C7×AGL(1,𝔽11) ⋊ C3
HallΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC2×ΓL(1,𝔽8)×D11 ⋊ C5
HallC7×C11⋊C5 ⋊ C22×C3
HallC2×C7×D11 ⋊ C3×C5
HallC11×ΓL(1,𝔽8) ⋊ C22×C5
HallC11⋊C5 ⋊ C22×ΓL(1,𝔽8)
HallC2×D11 ⋊ C5×ΓL(1,𝔽8)
SylowC11 ⋊ C22×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×AGL(1,𝔽11)

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×AGL(1,𝔽11)
commutator, cosocle, hypocenterC7×C11C22×C3×C5
FrattiniC1G
Fitting, socleC2×C7×C11C2×C3×C5
LCSG ▷ C7×C11
UCSC1 ◁ C2

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 154

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×ΓL(1,𝔽8)×AGL(1,𝔽11), C2×C3×ΓL(1,𝔽8)×AGL(1,𝔽11), C2×ΓL(1,𝔽8)×C3⋊AGL(1,𝔽11), …
maximal normal subgroup ofC2×C3×ΓL(1,𝔽8)×AGL(1,𝔽11), C2×C5×ΓL(1,𝔽8)×AGL(1,𝔽11), C2×C7×ΓL(1,𝔽8)×AGL(1,𝔽11), …
maximal quotient ofC2×C7⋊C9×AGL(1,𝔽11), D3×ΓL(1,𝔽8)×AGL(1,𝔽11), C2×C3×ΓL(1,𝔽8)×AGL(1,𝔽11), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4620, 41);