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ΓL(1,𝔽8)⋊AGL(1,𝔽31) #34086169

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

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊AGL(1,𝔽31)
unambiguous semidirect product((C31⋊C3)⋊C5)⋊AGL(1,𝔽7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊(D31⋊C3))⋊C5
unambiguous semidirect product(((C31⋊C3)⋊C5)⋊D7)⋊C3
unambiguous semidirect product(ΓL(1,𝔽8)⋊(D31⋊C5))⋊C3
unambiguous semidirect product(#13049961⋊C5)⋊C3
unambiguous semidirect product(#13049960⋊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
HallD217 ⋊ C32×C5
HallΓL(1,𝔽32) ⋊ C3×AGL(1,𝔽7)
SylowC31 ⋊ C3×C5×AGL(1,𝔽7)
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

Polycyclic presentation:

2,3^2,5,7,31|e^a=e^6,f^a=f^30,f^b=f^5,e^c=e^2,f^d=f^16

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 74);