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C5×D17×AGL(1,𝔽7) #20297397

order7140
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D17×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C5×D17)
unambiguous semidirect product(D7×D17)⋊(C3×C5)
unambiguous semidirect product(ΓL(1,𝔽8)×D17)⋊(C2×C5)
unambiguous semidirect product(C17×AGL(1,𝔽7))⋊(C2×C5)
unambiguous semidirect product(C17⋊AGL(1,𝔽7))⋊(C2×C5)
unambiguous semidirect product(C5×D7×D17)⋊C3

Splittings

Fitting, socle, HallC5×C7×C17 ⋊ C22×C3
derived, HallC7×C17 ⋊ C22×C3×C5
HallC5×D7×D17 ⋊ C3
HallC5×C17×ΓL(1,𝔽8) ⋊ C22
HallD7×D17 ⋊ C3×C5
HallC17×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×ΓL(1,𝔽8) ⋊ C2×D17
HallC5×C17 ⋊ C2×AGL(1,𝔽7)
HallC5×C7 ⋊ C2×C3×D17
HallΓL(1,𝔽8) ⋊ C2×C5×D17
SylowC17 ⋊ C2×C5×AGL(1,𝔽7)
SylowC7 ⋊ C2×C3×C5×D17

Subgroups

subgroupG/H
center, hypercenterC5D17×AGL(1,𝔽7)
commutator, hypocenterC7×C17C22×C3×C5
FrattiniC1G
Fitting, socleC5×C7×C17C22×C3
LCSG ▷ C7×C17
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C5×D17×AGL(1,𝔽7), C5×D17×C3⋊AGL(1,𝔽7), C5×AGL(1,𝔽7)×D51, …
maximal normal subgroup ofC3×C5×D17×AGL(1,𝔽7), C25×D17×AGL(1,𝔽7), C52×D17×AGL(1,𝔽7)
maximal quotient ofC5×D17×D7⋊C9, C3×C5×D17×AGL(1,𝔽7), C5×D17×C3⋊AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7140, 27);