Finite Group Databasebeta

← Prev (#44568566) Next (#44568568) →

C2×C7×ΓL(1,𝔽8)×AGL(1,𝔽11) #44568567

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

Aliases

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

Splittings

Fitting, socleC2×C72×C11 ⋊ C2×C3×C5
derivedC7×C11 ⋊ C22×C3×C5×C7
HallC2×C72×AGL(1,𝔽11) ⋊ C3
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C22
HallC2×C7×ΓL(1,𝔽8)×D11 ⋊ C5
HallC72×C11⋊C5 ⋊ C22×C3
HallC2×C72×D11 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC72×C11 ⋊ C22×C3×C5
HallC11⋊C5 ⋊ C22×C7×ΓL(1,𝔽8)
SylowC72 ⋊ C2×C3×AGL(1,𝔽11)
HallC2×D11 ⋊ C5×C7×ΓL(1,𝔽8)
SylowC11 ⋊ C22×C5×C7×ΓL(1,𝔽8)

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

200 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 462

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 269);