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ΓL(1,𝔽8)×(C19⋊C3)⋊AGL(1,𝔽5) #37561194

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

Aliases

direct productΓL(1,𝔽8)×(C19⋊C3)⋊AGL(1,𝔽5)
unambiguous semidirect product(C19⋊C3)⋊(AGL(1,𝔽5)×ΓL(1,𝔽8))
unambiguous semidirect product(C7×(C19⋊C3)⋊AGL(1,𝔽5))⋊C3
unambiguous semidirect product#22547638⋊C3
unambiguous semidirect product#22547639⋊C3
unambiguous semidirect product(ΓL(1,𝔽8)×C19⋊AGL(1,𝔽5))⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C19 ⋊ C32×C4
HallC5×ΓL(1,𝔽8)×C19⋊C3 ⋊ C4
HallC7×C19⋊AGL(1,𝔽5) ⋊ C32
HallΓL(1,𝔽8)×C19⋊C3 ⋊ AGL(1,𝔽5)
HallC19⋊AGL(1,𝔽5) ⋊ C3×ΓL(1,𝔽8)
HallC7×C19 ⋊ C32×AGL(1,𝔽5)
HallC5×C19 ⋊ C3×C4×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×(C19⋊C3)⋊C4
SylowC19 ⋊ C3×AGL(1,𝔽5)×ΓL(1,𝔽8)
SylowC7 ⋊ C3×(C19⋊C3)⋊AGL(1,𝔽5)
SylowC5 ⋊ ΓL(1,𝔽8)×(C19⋊C3)⋊C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C19C32×C4
prime radicalD5×ΓL(1,𝔽8)×C19⋊C3C2
cosocleC7×D5×C19C2×C32
LCSG ▷ C5×C7×C19
UCSC1

144 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 41 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) = 665

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 326);