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C3⋊(ΓL(1,𝔽8)⋊(D31⋊C5)) #34086191

order19530
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(ΓL(1,𝔽8)⋊(D31⋊C5))
unambiguous semidirect productΓL(1,𝔽8)⋊(C3⋊(D31⋊C5))
unambiguous semidirect product(C3⋊(ΓL(1,𝔽8)⋊D31))⋊C5

Splittings

Fitting, socle, derivedC3×C7×C31 ⋊ C2×C3×C5
HallC3×ΓL(1,𝔽8)×ΓL(1,𝔽32) ⋊ C2
HallC3⋊(ΓL(1,𝔽8)⋊D31) ⋊ C5
HallC3×ΓL(1,𝔽8)×C31 ⋊ C2×C5
HallC7×ΓL(1,𝔽32) ⋊ C3×D3
HallC7×C31 ⋊ C3×C5×D3
HallΓL(1,𝔽32) ⋊ C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ D31⋊C5
SylowC31 ⋊ C5×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C3×C3⋊(D31⋊C5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C31C2×C3×C5
LCSG ▷ C3×C7×C31
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 651

Polycyclic presentation:

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

Conjugacy classes & character table

105 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 96);