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C5×C8×ΓL(1,𝔽8) #11719702

order840
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C8 · Schur multiplier (H2) = C1

Aliases

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

Splittings

Fitting, HallC5×C7×C8 ⋊ C3
derived, SylowC7 ⋊ C3×C5×C8
HallC7×C8 ⋊ C3×C5
HallC5×C7 ⋊ C3×C8

Subgroups

subgroupG/H
center, hypercenterC5×C8ΓL(1,𝔽8)
commutator, hypocenterC7C3×C5×C8
FrattiniC4C2×C5×ΓL(1,𝔽8)
FittingC5×C7×C8C3
socleC2×C5×C7C3×C4
prime radicalC2×C5×ΓL(1,𝔽8)C4
cosocleC4×C7C2×C3×C5
LCSG ▷ C7
UCSC1 ◁ C5×C8

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×ΓL(1,𝔽8)⋊C16, ΓL(1,𝔽8)×#672, ΓL(1,𝔽8)⋊#672, …
maximal normal subgroup ofC5×ΓL(1,𝔽8)⋊C16, ΓL(1,𝔽8)×#672, ΓL(1,𝔽8)⋊#672, …
maximal quotient ofC5×C16×ΓL(1,𝔽8), C2×C5×C8×ΓL(1,𝔽8), C5×ΓL(1,𝔽8)×AGL(1,𝔽9)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 4);