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C3×D5×C17×ΓL(1,𝔽8) #24924216

order10710
automorphism?
homologyAbelianization (H1) = C2×C32×C17 · Schur multiplier (H2) = C3

Aliases

direct productC3×D5×C17×ΓL(1,𝔽8)
unambiguous semidirect productC5⋊(C2×C3×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D5)⋊(C32×C17)
unambiguous semidirect product(C3×C7×D5)⋊(C3×C17)
unambiguous semidirect product(C7×D5×C17)⋊C32
unambiguous semidirect product(C3×C7×D5×C17)⋊C3

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC3×C17D5×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC5×C7C2×C32×C17
FrattiniC1G
Fitting, socleC3×C5×C7×C17C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C3×C17

96 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1020 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C17×ΓL(1,𝔽8)⋊AGL(1,𝔽5), C3×ΓL(1,𝔽8)⋊#66774, C3×C17×AGL(1,𝔽5)×ΓL(1,𝔽8), …
maximal normal subgroup ofC3×C17×ΓL(1,𝔽8)⋊AGL(1,𝔽5), C3×ΓL(1,𝔽8)⋊#66774, C3×C17×AGL(1,𝔽5)×ΓL(1,𝔽8), …
maximal quotient ofC3×C17×Dic5×ΓL(1,𝔽8), C2×C3×D5×C17×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 31);