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C5×D7×ΓL(1,𝔽8)×D11 #44568632

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

Aliases

direct productC5×D7×ΓL(1,𝔽8)×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
HallC7×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 ⋊ C2×D7×ΓL(1,𝔽8)
SylowC72 ⋊ C2×C3×C5×D11
SylowC11 ⋊ C2×C5×D7×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC5D7×ΓL(1,𝔽8)×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 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) = 2310

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 334);