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C4×C5×ΓL(1,𝔽8)×C19⋊C3 #37561089

order23940
automorphism?
homologyAbelianization (H1) = C32×C4×C5 · Schur multiplier (H2) = C3

Aliases

direct productC4×C5×ΓL(1,𝔽8)×C19⋊C3
unambiguous semidirect product(C4×C7×C19⋊C3)⋊(C3×C5)
unambiguous semidirect product(C4×C19×ΓL(1,𝔽8))⋊(C3×C5)
unambiguous semidirect product(C4×#87704)⋊(C3×C5)
unambiguous semidirect product(C4×#87705)⋊(C3×C5)
unambiguous semidirect product(C4×C5×C7×C19⋊C3)⋊C3
unambiguous semidirect product(C4×C5×C19×ΓL(1,𝔽8))⋊C3
unambiguous semidirect product(C4×C5×#87704)⋊C3
unambiguous semidirect product(C4×C5×#87705)⋊C3

Splittings

Fitting, HallC4×C5×C7×C19 ⋊ C32
derived, HallC7×C19 ⋊ C32×C4×C5
HallC5×C7×C19 ⋊ C32×C4
HallC4×C7×C19 ⋊ C32×C5
HallC4×C5×C19 ⋊ C3×ΓL(1,𝔽8)
HallC4×C5×C7 ⋊ C3×C19⋊C3
HallC5×C19 ⋊ C3×C4×ΓL(1,𝔽8)
HallC4×C19 ⋊ C3×C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×C4×C19⋊C3
HallC4×C7 ⋊ C3×C5×C19⋊C3
SylowC19 ⋊ C3×C4×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C4×C5×C19⋊C3

Subgroups

subgroupG/H
center, hypercenterC4×C5ΓL(1,𝔽8)×C19⋊C3
commutator, hypocenterC7×C19C32×C4×C5
FrattiniC2C2×C5×ΓL(1,𝔽8)×C19⋊C3
FittingC4×C5×C7×C19C32
socleC2×C5×C7×C19C2×C32
prime radicalC2×C5×ΓL(1,𝔽8)×C19⋊C3C2
cosocleC2×C7×C19C2×C32×C5
LCSG ▷ C7×C19
UCSC1 ◁ C4×C5

144 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 2660

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 221);