Finite Group Databasebeta

← Prev (#45654455) Next (#45654457) →

C5⋊(ΓL(1,𝔽8)⋊(D31⋊C5)) #45654456

order32550
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(ΓL(1,𝔽8)⋊(D31⋊C5))
unambiguous semidirect productΓL(1,𝔽8)⋊(C5⋊(D31⋊C5))
unambiguous semidirect product(C5⋊(ΓL(1,𝔽8)⋊D31))⋊C5

Splittings

Fitting, socle, derivedC5×C7×C31 ⋊ C2×C3×C5
HallC5×ΓL(1,𝔽8)×ΓL(1,𝔽32) ⋊ C2
HallC5⋊(C7⋊(D31⋊C5)) ⋊ C3
HallC5×C7×ΓL(1,𝔽32) ⋊ C2×C3
HallC5×ΓL(1,𝔽32) ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8)×C31 ⋊ C5×D5
HallC7×C31 ⋊ C3×C5×D5
SylowC31 ⋊ C5×C5⋊AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C5⋊(D31⋊C5)
SylowC7 ⋊ C3×C5⋊(D31⋊C5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C31C2×C3×C5
LCSG ▷ C5×C7×C31
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1085

Polycyclic presentation:

2,3,5^2,7,31|d^a=d^4,e^a=e^6,f^a=f^30,e^b=e^2,f^c=f^16

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32550, 92);