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C5×(ΓL(1,𝔽8)⋊D7)⋊D11 #44568631

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

Aliases

direct productC5×(ΓL(1,𝔽8)⋊D7)⋊D11
unambiguous semidirect product(ΓL(1,𝔽8)⋊D7)⋊(C5×D11)

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C22×C3
derived, HallC72×C11 ⋊ C22×C3×C5
HallC5×(C7⋊D7)⋊D11 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C22
Hall(C7⋊D7)⋊D11 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×C7×ΓL(1,𝔽8) ⋊ C2×D11
HallC5×C72 ⋊ C2×C3×D11
HallC7×ΓL(1,𝔽8) ⋊ C2×C5×D11
HallC5×C11 ⋊ D7×AGL(1,𝔽7)
SylowC72 ⋊ C2×C3×C5×D11
SylowC11 ⋊ C5×D7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC5(ΓL(1,𝔽8)⋊D7)⋊D11
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C72×C11
UCSC1 ◁ C5

168 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 2310

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 333);