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C2×C5×C11×ΓL(1,𝔽8)⋊D7 #44568765

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5×C11 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×C11×ΓL(1,𝔽8)⋊D7
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C5×C11×D7)

Splittings

Fitting, socleC2×C5×C72×C11 ⋊ C2×C3
derived, SylowC72 ⋊ C22×C3×C5×C11
HallC2×C5×C11×C7⋊D7 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C22
HallC5×C72×C11 ⋊ C22×C3
HallC2×C11×C7⋊D7 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC2×C5×C7⋊D7 ⋊ C3×C11
HallC5×C7×ΓL(1,𝔽8) ⋊ C22×C11
HallC72×C11 ⋊ C22×C3×C5
HallC5×C72 ⋊ C22×C3×C11
HallC2×C7⋊D7 ⋊ C3×C5×C11
HallC7×ΓL(1,𝔽8) ⋊ C22×C5×C11

Subgroups

subgroupG/H
center, hypercenterC2×C5×C11ΓL(1,𝔽8)⋊D7
commutator, cosocle, hypocenterC72C22×C3×C5×C11
FrattiniC1G
Fitting, socleC2×C5×C72×C11C2×C3
LCSG ▷ C72
UCSC1 ◁ C2×C5×C11

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) = 2310

Conjugacy classes & character table

2420 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 467);