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C5×ΓL(1,𝔽8)⋊C8 #11719699

order840
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C5×C8 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)⋊C8
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C8)
unambiguous semidirect product(C7⋊C8)⋊(C3×C5)
unambiguous semidirect product(C5×C7⋊C8)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C5×C8
HallC5×C7⋊C8 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C8
HallC7⋊C8 ⋊ C3×C5
HallC5×C7 ⋊ C3×C8
HallΓL(1,𝔽8) ⋊ C5×C8

Subgroups

subgroupG/H
center, hypercenterC4×C5AGL(1,𝔽7)
commutator, hypocenterC7C3×C5×C8
FrattiniC4C5×AGL(1,𝔽7)
FittingC4×C5×C7C2×C3
socleC2×C5×C7C3×C4
prime radicalC2×C5×ΓL(1,𝔽8)C4
cosocleC4×C7C2×C3×C5
LCSG ▷ C7
UCSC1 ◁ C4×C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×ΓL(1,𝔽8)⋊C8, #10587837⋊C3, #10587839⋊C3, …
maximal normal subgroup ofD5×ΓL(1,𝔽8)⋊C8, #10587837⋊C3, #10587839⋊C3, …
maximal quotient ofC5×ΓL(1,𝔽8)⋊C16, C2×C5×ΓL(1,𝔽8)⋊C8, C5×ΓL(1,𝔽8)⋊AGL(1,𝔽9)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 1);