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C5×D3×C17×ΓL(1,𝔽8) #24924220

order10710
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C17 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3×C17×ΓL(1,𝔽8)
unambiguous semidirect productC3⋊(C2×C5×C17×ΓL(1,𝔽8))
unambiguous semidirect product(D3×C7)⋊(C3×C5×C17)
unambiguous semidirect product(C5×D3×C7)⋊(C3×C17)
unambiguous semidirect product(D3×C7×C17)⋊(C3×C5)
unambiguous semidirect product(C5×D3×C7×C17)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C2×C3
derivedC3×C7 ⋊ C2×C3×C5×C17
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C2
HallC3×C17×ΓL(1,𝔽8) ⋊ C2×C5
HallC5×C7×C17 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×C17
HallC7×C17 ⋊ C3×C5×D3
HallC3×ΓL(1,𝔽8) ⋊ C2×C5×C17
HallC5×C7 ⋊ C3×D3×C17
SylowC7 ⋊ C3×C5×D3×C17

Subgroups

subgroupG/H
center, hypercenterC5×C17D3×ΓL(1,𝔽8)
commutator, cosocle, hypocenterC3×C7C2×C3×C5×C17
FrattiniC1G
Fitting, socleC3×C5×C7×C17C2×C3
LCSG ▷ C3×C7
UCSC1 ◁ C5×C17

72 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1275 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC5×D3×C17×AGL(1,𝔽7), C5×D3×C17⋊AGL(1,𝔽7), C5×D3×ΓL(1,𝔽8)×D17, …
maximal normal subgroup ofC5×D3×C17×AGL(1,𝔽7), C5×D3×C17⋊AGL(1,𝔽7), C5×D3×ΓL(1,𝔽8)×D17, …
maximal quotient ofC5×Dic3×C17×ΓL(1,𝔽8), C2×C5×D3×C17×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 35);