Finite Group Databasebeta

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D5×ΓL(1,𝔽8)×D77 #44568711

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

Aliases

direct productD5×ΓL(1,𝔽8)×D77
unambiguous semidirect product(D5×C11)⋊(D7×ΓL(1,𝔽8))

Splittings

Fitting, socle, derived, HallC5×C72×C11 ⋊ C22×C3
HallC7×D5×D77 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C22
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×D5
HallC5×C7×ΓL(1,𝔽8) ⋊ C2×D11
HallC72×C11 ⋊ C2×C3×D5
HallC5×C72 ⋊ C2×C3×D11
HallC7×ΓL(1,𝔽8) ⋊ D5×D11
HallC5×C11 ⋊ C2×D7×ΓL(1,𝔽8)
SylowC72 ⋊ C3×D5×D11
SylowC11 ⋊ D5×D7×ΓL(1,𝔽8)
SylowC5 ⋊ C2×ΓL(1,𝔽8)×D77

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C72×C11C22×C3
LCSG ▷ C5×C72×C11
UCSC1

168 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 48 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

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 413);