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C2×C3×C5×C17×ΓL(1,𝔽8) #24924226

order10710
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C32×C5×C17 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×C5×C17×ΓL(1,𝔽8)
unambiguous semidirect product(C2×C7)⋊(C32×C5×C17)
unambiguous semidirect product(C2×C3×C7)⋊(C3×C5×C17)
unambiguous semidirect product(C2×C5×C7)⋊(C32×C17)
unambiguous semidirect product(C2×C3×C5×C7)⋊(C3×C17)
unambiguous semidirect product(C2×C7×C17)⋊(C32×C5)
unambiguous semidirect product(C2×C3×C7×C17)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C7×C17)⋊C32
unambiguous semidirect product(C2×C3×C5×C7×C17)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7×C17 ⋊ C3
derived, SylowC7 ⋊ C2×C32×C5×C17
HallC2×C5×C7×C17 ⋊ C32
HallC5×C7×C17 ⋊ C2×C32
HallC2×C7×C17 ⋊ C32×C5
HallC7×C17 ⋊ C2×C32×C5
HallC2×C5×C7 ⋊ C32×C17
HallC5×C7 ⋊ C2×C32×C17
HallC2×C7 ⋊ C32×C5×C17

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5×C17ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C32×C5×C17
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C17C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C5×C17

96 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2550 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×C5×C17×ΓL(1,𝔽8)⋊C4, C3×C5×ΓL(1,𝔽8)⋊Dic17, C3×C5×ΓL(1,𝔽8)×Dic17, …
maximal normal subgroup ofC3×C5×C17×ΓL(1,𝔽8)⋊C4, C3×C5×ΓL(1,𝔽8)⋊Dic17, C3×C5×ΓL(1,𝔽8)×Dic17, …
maximal quotient ofC3×C4×C5×C17×ΓL(1,𝔽8), C22×C3×C5×C17×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 41);