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C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7)) #17984391

order4410
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C5⋊(ΓL(1,𝔽8)⋊D7))
unambiguous semidirect productC5⋊(C3⋊(ΓL(1,𝔽8)⋊D7))
unambiguous semidirect productΓL(1,𝔽8)⋊D105

Splittings

Fitting, socle, derivedC3×C5×C72 ⋊ C2×C3
HallC3×C5×C7×ΓL(1,𝔽8) ⋊ C2
HallC3×C7×ΓL(1,𝔽8) ⋊ D5
HallC5×C72 ⋊ C3×D3
SylowC72 ⋊ C3×D15
SylowC5 ⋊ C3⋊(ΓL(1,𝔽8)⋊D7)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C72C2×C3
LCSG ▷ C3×C5×C72
UCSC1

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 315

Polycyclic presentation:

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

Conjugacy classes & character table

267 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3⋊(ΓL(1,𝔽8)⋊D175), C5×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7)), C3⋊(C5⋊(C5⋊(ΓL(1,𝔽8)⋊D7)))
maximal normal subgroup ofC5×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7))
maximal quotient ofC3⋊(ΓL(1,𝔽8)⋊D175), C5×C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7)), C3⋊(C5⋊(C5⋊(ΓL(1,𝔽8)⋊D7)))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4410, 112);