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ΓL(1,𝔽8)×AGL(1,𝔽31) #34086171

order19530
automorphism?
homologyAbelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productΓL(1,𝔽8)×AGL(1,𝔽31)
unambiguous semidirect product(D31⋊C3)⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product((C31⋊C3)⋊C5)⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(ΓL(1,𝔽8)×D31⋊C3)⋊C5
unambiguous semidirect product(C7×AGL(1,𝔽31))⋊C3
unambiguous semidirect product(ΓL(1,𝔽8)×D31⋊C5)⋊C3
unambiguous semidirect product(#13049955⋊C5)⋊C3
unambiguous semidirect product(#13049954⋊C5)⋊C3

Splittings

Fitting, socle, derived, HallC7×C31 ⋊ C2×C32×C5
HallΓL(1,𝔽8)×(C31⋊C3)⋊C5 ⋊ C2
HallΓL(1,𝔽8)×D31⋊C3 ⋊ C5
HallC7×D31⋊C5 ⋊ C32
HallΓL(1,𝔽8)×C31⋊C3 ⋊ C2×C5
HallC7×ΓL(1,𝔽32) ⋊ C2×C32
HallC7×D31 ⋊ C32×C5
HallD31⋊C5 ⋊ C3×ΓL(1,𝔽8)
HallΓL(1,𝔽32) ⋊ C2×C3×ΓL(1,𝔽8)
HallD31 ⋊ C3×C5×ΓL(1,𝔽8)
SylowC31 ⋊ C2×C3×C5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×AGL(1,𝔽31)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C31C2×C32×C5
LCSG ▷ C7×C31
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 217

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 76);