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C3×ΓL(1,𝔽8)×D85 #24924217

order10710
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×D85
unambiguous semidirect productC5⋊(C3×ΓL(1,𝔽8)×D17)
unambiguous semidirect productC17⋊(C3×D5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D85)⋊C32
unambiguous semidirect product(C3×C7×D85)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C2×C3
derived, HallC5×C7×C17 ⋊ C2×C32
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C2
HallC7×D85 ⋊ C32
HallC3×C17×ΓL(1,𝔽8) ⋊ D5
HallC3×C5×ΓL(1,𝔽8) ⋊ D17
HallC7×C17 ⋊ C32×D5
HallC5×C17 ⋊ C2×C3×ΓL(1,𝔽8)
HallC5×C7 ⋊ C32×D17
SylowC17 ⋊ C3×D5×ΓL(1,𝔽8)
SylowC7 ⋊ C32×D85
SylowC5 ⋊ C3×ΓL(1,𝔽8)×D17

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×D85
commutator, cosocle, hypocenterC5×C7×C17C2×C32
FrattiniC1G
Fitting, socleC3×C5×C7×C17C2×C3
LCSG ▷ C5×C7×C17
UCSC1 ◁ C3

96 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

660 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)⋊#66775, C3×ΓL(1,𝔽8)⋊#66776, C3×ΓL(1,𝔽8)×#66775, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)⋊#66775, C3×ΓL(1,𝔽8)⋊#66776, C3×ΓL(1,𝔽8)×#66775, …
maximal quotient ofC3×ΓL(1,𝔽8)×Dic85, C2×C3×ΓL(1,𝔽8)×D85

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 32);