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C2×C7×ΓL(1,𝔽8)⋊AGL(1,𝔽11) #44568568

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5×C7 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×ΓL(1,𝔽8)⋊AGL(1,𝔽11)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C7×AGL(1,𝔽11))
unambiguous semidirect product(C2×C11⋊C5)⋊(C7×AGL(1,𝔽7))
unambiguous semidirect product(C2×C11⋊AGL(1,𝔽7))⋊(C5×C7)
unambiguous semidirect product(C2×C7⋊AGL(1,𝔽11))⋊(C3×C7)
unambiguous semidirect product(C2×C7×C11⋊AGL(1,𝔽7))⋊C5

Splittings

Fitting, socleC2×C72×C11 ⋊ C2×C3×C5
derivedC7×C11 ⋊ C22×C3×C5×C7
HallC2×C7×C7⋊AGL(1,𝔽11) ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC2×C7×C11⋊AGL(1,𝔽7) ⋊ C5
HallC72×C11⋊C5 ⋊ C22×C3
HallC2×C7×D77 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC72×C11 ⋊ C22×C3×C5
HallC7×ΓL(1,𝔽8) ⋊ C2×AGL(1,𝔽11)
HallC11⋊C5 ⋊ C2×C7×AGL(1,𝔽7)
SylowC72 ⋊ C2×C3×AGL(1,𝔽11)
SylowC11 ⋊ C2×C5×C7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC2×C7ΓL(1,𝔽8)⋊AGL(1,𝔽11)
commutator, cosocle, hypocenterC7×C11C22×C3×C5×C7
FrattiniC1G
Fitting, socleC2×C72×C11C2×C3×C5
LCSG ▷ C7×C11
UCSC1 ◁ C2×C7

168 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 924

Conjugacy classes & character table

560 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 270);