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C2×C5×ΓL(1,𝔽8)×D19⋊C3 #37561231

order23940
automorphism?
homologyAbelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C5×ΓL(1,𝔽8)×D19⋊C3
unambiguous semidirect product(C19⋊C3)⋊(C22×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C19⋊C3)⋊(C2×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D19⋊C3)⋊(C3×C5)
unambiguous semidirect product(C2×ΓL(1,𝔽8)×D19)⋊(C3×C5)
unambiguous semidirect product(C2×#11716321)⋊(C3×C5)
unambiguous semidirect product(C2×#11716322)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C7×D19⋊C3)⋊C3
unambiguous semidirect product(C2×C5×#11716321)⋊C3
unambiguous semidirect product(C2×C5×#11716322)⋊C3
unambiguous semidirect product(C2×C5×ΓL(1,𝔽8)×D19)⋊C3

Splittings

Fitting, socleC2×C5×C7×C19 ⋊ C2×C32
derived, HallC7×C19 ⋊ C22×C32×C5
HallC5×ΓL(1,𝔽8)×C19⋊C3 ⋊ C22
HallC2×C5×C7×D19 ⋊ C32
HallΓL(1,𝔽8)×C19⋊C3 ⋊ C22×C5
HallC5×C7×C19 ⋊ C22×C32
HallC2×C7×D19 ⋊ C32×C5
HallC2×C5×D19 ⋊ C3×ΓL(1,𝔽8)
HallC5×C19 ⋊ C22×C3×ΓL(1,𝔽8)
HallC2×D19 ⋊ C3×C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C3×D19⋊C3
SylowC19 ⋊ C22×C3×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×C5×D19⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)×D19⋊C3
commutator, cosocle, hypocenterC7×C19C22×C32×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C19C2×C32
LCSG ▷ C7×C19
UCSC1 ◁ C2×C5

240 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 1330

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 363);