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C5×ΓL(1,𝔽8)×C83 #22576550

order8715
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C83 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)×C83
unambiguous semidirect productC7⋊(C3×C5×C83)
unambiguous semidirect product(C5×C7)⋊(C3×C83)
unambiguous semidirect product(C7×C83)⋊(C3×C5)
unambiguous semidirect product(C5×C7×C83)⋊C3

Splittings

Fitting, socle, HallC5×C7×C83 ⋊ C3
derived, SylowC7 ⋊ C3×C5×C83
HallC7×C83 ⋊ C3×C5
HallC5×C7 ⋊ C3×C83

Subgroups

subgroupG/H
center, hypercenterC5×C83ΓL(1,𝔽8)
commutator, hypocenterC7C3×C5×C83
FrattiniC1G
Fitting, socleC5×C7×C83C3
LCSG ▷ C7
UCSC1 ◁ C5×C83

16 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2075 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C5×ΓL(1,𝔽8)×C83, C5×ΓL(1,𝔽8)×D83, C5×AGL(1,𝔽7)×C83, …
maximal normal subgroup ofC2×C5×ΓL(1,𝔽8)×C83, C5×ΓL(1,𝔽8)×D83, C5×AGL(1,𝔽7)×C83, …
maximal quotient ofC2×C5×ΓL(1,𝔽8)×C83, C5×C7⋊C9×C83, C3×C5×ΓL(1,𝔽8)×C83, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8715, 2);