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C2×C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7)) #44568786

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

Aliases

direct productC2×C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7))
unambiguous semidirect product(C2×C5)⋊(C11⋊(ΓL(1,𝔽8)⋊D7))
unambiguous semidirect product(C2×C11)⋊(C5⋊(ΓL(1,𝔽8)⋊D7))
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D385

Splittings

Fitting, socleC2×C5×C72×C11 ⋊ C2×C3
derived, HallC5×C72×C11 ⋊ C22×C3
HallC2×C5⋊(C7⋊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) ⋊ C2×D55
HallC5×C11 ⋊ C2×ΓL(1,𝔽8)⋊D7
SylowC72 ⋊ C2×C3×D55
SylowC11 ⋊ C2×C5⋊(ΓL(1,𝔽8)⋊D7)
SylowC5 ⋊ C2×C11⋊(ΓL(1,𝔽8)⋊D7)

Subgroups

subgroupG/H
center, hypercenterC2C5⋊(C11⋊(ΓL(1,𝔽8)⋊D7))
commutator, cosocle, hypocenterC5×C72×C11C22×C3
FrattiniC1G
Fitting, socleC2×C5×C72×C11C2×C3
LCSG ▷ C5×C72×C11
UCSC1 ◁ C2

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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

1934 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 488);