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AGL(1,𝔽7)×ΓL(1,𝔽32) #20266751

order6510
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productAGL(1,𝔽7)×ΓL(1,𝔽32)
unambiguous semidirect productD7⋊(C3×ΓL(1,𝔽32))
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×ΓL(1,𝔽32))
unambiguous semidirect product(C31×AGL(1,𝔽7))⋊C5

Splittings

Fitting, socle, derived, HallC7×C31 ⋊ C2×C3×C5
HallΓL(1,𝔽8)×ΓL(1,𝔽32) ⋊ C2
HallD7×ΓL(1,𝔽32) ⋊ C3
HallC31×AGL(1,𝔽7) ⋊ C5
HallC7×ΓL(1,𝔽32) ⋊ C2×C3
HallΓL(1,𝔽8)×C31 ⋊ C2×C5
HallD7×C31 ⋊ C3×C5
SylowC31 ⋊ C5×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ C2×ΓL(1,𝔽32)
HallD7 ⋊ C3×ΓL(1,𝔽32)
SylowC7 ⋊ C2×C3×ΓL(1,𝔽32)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC7×C31C2×C3×C5
LCSG ▷ C7×C31
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D31⋊C5, C2×AGL(1,𝔽7)×ΓL(1,𝔽32), AGL(1,𝔽7)×(C31⋊C3)⋊C5, …
maximal normal subgroup ofAGL(1,𝔽7)×D31⋊C5, C2×AGL(1,𝔽7)×ΓL(1,𝔽32), AGL(1,𝔽7)×(C31⋊C3)⋊C5, …
maximal quotient ofΓL(1,𝔽8)⋊C4×ΓL(1,𝔽32), C2×AGL(1,𝔽7)×ΓL(1,𝔽32), D7⋊C9×ΓL(1,𝔽32), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6510, 66);