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C2×D5×ΓL(1,𝔽8)×C13⋊C3 #30655309

order16380
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×D5×ΓL(1,𝔽8)×C13⋊C3
unambiguous semidirect productC5⋊(C22×ΓL(1,𝔽8)×C13⋊C3)
unambiguous semidirect product(C2×C5)⋊(C2×ΓL(1,𝔽8)×C13⋊C3)
unambiguous semidirect product(C2×C7×D5×C13⋊C3)⋊C3
unambiguous semidirect product(C2×D5×C13×ΓL(1,𝔽8))⋊C3
unambiguous semidirect product(C2×D5×#63270)⋊C3
unambiguous semidirect product(C2×D5×#63271)⋊C3

Splittings

Fitting, socleC2×C5×C7×C13 ⋊ C2×C32
derived, HallC5×C7×C13 ⋊ C22×C32
HallC5×ΓL(1,𝔽8)×C13⋊C3 ⋊ C22
HallC2×C7×D5×C13 ⋊ C32
HallC2×D5×C13 ⋊ C3×ΓL(1,𝔽8)
HallC2×C7×D5 ⋊ C3×C13⋊C3
HallC7×C13 ⋊ C2×C32×D5
HallC5×C13 ⋊ C22×C3×ΓL(1,𝔽8)
HallC5×C7 ⋊ C22×C3×C13⋊C3
SylowC13 ⋊ C2×C3×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×D5×C13⋊C3
SylowC5 ⋊ C22×ΓL(1,𝔽8)×C13⋊C3

Subgroups

subgroupG/H
center, hypercenterC2D5×ΓL(1,𝔽8)×C13⋊C3
commutator, cosocle, hypocenterC5×C7×C13C22×C32
FrattiniC1G
Fitting, socleC2×C5×C7×C13C2×C32
LCSG ▷ C5×C7×C13
UCSC1 ◁ C2

240 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 910

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16380, 359);