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C5×C3⋊(ΓL(1,𝔽8)⋊D59) #49151209

order37170
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(ΓL(1,𝔽8)⋊D59)
unambiguous semidirect productC3⋊(C5×ΓL(1,𝔽8)⋊D59)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D177)
unambiguous semidirect productC59⋊(C5×C3⋊AGL(1,𝔽7))
unambiguous semidirect productD1239⋊(C3×C5)
unambiguous semidirect product(C5×D1239)⋊C3

Splittings

Fitting, socleC3×C5×C7×C59 ⋊ C2×C3
derivedC3×C7×C59 ⋊ C2×C3×C5
HallC3×C5×ΓL(1,𝔽8)×C59 ⋊ C2
HallC3×ΓL(1,𝔽8)×C59 ⋊ C2×C5
HallC5×C7×C59 ⋊ C3×D3
HallC7×C59 ⋊ C3×C5×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D59
HallC5×C59 ⋊ C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C5×D59
SylowC59 ⋊ C5×C3⋊AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D177
SylowC7 ⋊ C3×C5×D177

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(ΓL(1,𝔽8)⋊D59)
commutator, cosocle, hypocenterC3×C7×C59C2×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C59C2×C3
LCSG ▷ C3×C7×C59
UCSC1 ◁ C5

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 6195

Conjugacy classes & character table

2235 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37170, 34);