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C19×ΓL(1,𝔽8)×ΓL(1,𝔽16) #37561326

order23940
automorphism?
homologyAbelianization (H1) = C3×C4×C19 · Schur multiplier (H2) = C1

Aliases

direct productC19×ΓL(1,𝔽8)×ΓL(1,𝔽16)
unambiguous semidirect productC3⋊(C19×AGL(1,𝔽5)×ΓL(1,𝔽8))
unambiguous semidirect product(C7×ΓL(1,𝔽16))⋊(C3×C19)

Splittings

Fitting, socleC3×C5×C7×C19 ⋊ C3×C4
derivedC3×C5×C7 ⋊ C3×C4×C19
HallC3×C5×C19×ΓL(1,𝔽8) ⋊ C4
HallC3×C19×ΓL(1,𝔽8) ⋊ AGL(1,𝔽5)
HallC5×C7×C19 ⋊ C3×Dic3
HallC3×C5×ΓL(1,𝔽8) ⋊ C4×C19
HallC7×C19 ⋊ C3×ΓL(1,𝔽16)
HallC5×C19 ⋊ Dic3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ C19×AGL(1,𝔽5)
HallC5×C7 ⋊ C3×Dic3×C19
SylowC7 ⋊ C3×C19×ΓL(1,𝔽16)
SylowC5 ⋊ Dic3×C19×ΓL(1,𝔽8)

Subgroups

subgroupG/H
center, hypercenterC19ΓL(1,𝔽8)×ΓL(1,𝔽16)
commutator, hypocenterC3×C5×C7C3×C4×C19
FrattiniC1G
Fitting, socleC3×C5×C7×C19C3×C4
prime radicalC3×D5×C19×ΓL(1,𝔽8)C2
cosocleC3×C7×D5C2×C3×C19
LCSG ▷ C3×C5×C7
UCSC1 ◁ C19

112 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 1995

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 458);