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C5×C8×AGL(1,𝔽7) #13132020

order1680
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C8 · Schur multiplier (H2) = C2

Aliases

direct productC5×C8×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C5×C8)
unambiguous semidirect product(C8×D7)⋊(C3×C5)
unambiguous semidirect product(C5×C8×D7)⋊C3

Splittings

FittingC5×C7×C8 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C5×C8
HallC5×C8×D7 ⋊ C3
HallC8×D7 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ C2×C8
HallC5×C7 ⋊ C2×C3×C8
HallΓL(1,𝔽8) ⋊ C2×C5×C8

Subgroups

subgroupG/H
center, hypercenterC5×C8AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C5×C8
FrattiniC4C2×C5×AGL(1,𝔽7)
FittingC5×C7×C8C2×C3
socleC2×C5×C7C2×C3×C4
prime radicalC2×C5×AGL(1,𝔽7)C4
cosocleC4×C7C22×C3×C5
LCSG ▷ C7
UCSC1 ◁ C5×C8

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC5×AGL(1,𝔽7)×AGL(1,𝔽9), C8×D5×AGL(1,𝔽7), C5×M16×AGL(1,𝔽7), …
maximal normal subgroup ofC8×D5×AGL(1,𝔽7), C5×M16×AGL(1,𝔽7), AGL(1,𝔽7)×#676, …
maximal quotient ofC5×AGL(1,𝔽7)×AGL(1,𝔽9)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 40);