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C3⋊(C5⋊(C17⋊AGL(1,𝔽7))) #24924223

order10710
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C5⋊(C17⋊AGL(1,𝔽7)))
unambiguous semidirect productC5⋊(C3⋊(C17⋊AGL(1,𝔽7)))
unambiguous semidirect productC17⋊(C3⋊(C5⋊AGL(1,𝔽7)))
unambiguous semidirect productΓL(1,𝔽8)⋊D255
unambiguous semidirect productD1785⋊C3

Splittings

Fitting, socle, derivedC3×C5×C7×C17 ⋊ C2×C3
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C2
HallC3×C17×ΓL(1,𝔽8) ⋊ D5
HallC5×C7×C17 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D17
HallC7×C17 ⋊ C3×D15
HallC5×C17 ⋊ C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ D85
HallC5×C7 ⋊ C3×D51
SylowC17 ⋊ C3⋊(C5⋊AGL(1,𝔽7))
SylowC7 ⋊ C3×D255
SylowC5 ⋊ C3⋊(C17⋊AGL(1,𝔽7))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C5×C7×C17C2×C3
LCSG ▷ C3×C5×C7×C17
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^2,5,7,17|c^a=c^2,d^a=d^4,e^a=e^6,f^a=f^16,e^b=e^2

Conjugacy classes & character table

642 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD17×C3⋊(C5⋊AGL(1,𝔽7)), AGL(1,𝔽7)×D255, D15×C17⋊AGL(1,𝔽7), …
maximal normal subgroup ofD17×C3⋊(C5⋊AGL(1,𝔽7)), AGL(1,𝔽7)×D255, D15×C17⋊AGL(1,𝔽7), …
maximal quotient ofC3⋊(ΓL(1,𝔽8)⋊Dic85), C2×C3⋊(C5⋊(C17⋊AGL(1,𝔽7)))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 38);