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C5×ΓL(1,𝔽8)×D51 #24924221

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

Aliases

direct productC5×ΓL(1,𝔽8)×D51
unambiguous semidirect productC3⋊(C5×ΓL(1,𝔽8)×D17)
unambiguous semidirect productC17⋊(C5×D3×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D51)⋊(C3×C5)
unambiguous semidirect product(C5×C7×D51)⋊C3

Splittings

Fitting, socleC3×C5×C7×C17 ⋊ C2×C3
derivedC3×C7×C17 ⋊ C2×C3×C5
HallC3×C5×C17×ΓL(1,𝔽8) ⋊ C2
HallC3×C17×ΓL(1,𝔽8) ⋊ C2×C5
HallC5×C7×C17 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D17
HallC7×C17 ⋊ C3×C5×D3
HallC5×C17 ⋊ D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ C5×D17
HallC5×C7 ⋊ C3×D51
SylowC17 ⋊ C5×D3×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C5×D51

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)×D51
commutator, cosocle, hypocenterC3×C7×C17C2×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C17C2×C3
LCSG ▷ C3×C7×C17
UCSC1 ◁ C5

72 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

675 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×AGL(1,𝔽7)×D51, C5×D51⋊AGL(1,𝔽7), C5×D3×ΓL(1,𝔽8)×D17, …
maximal normal subgroup ofC5×AGL(1,𝔽7)×D51, C5×D51⋊AGL(1,𝔽7), C5×D3×ΓL(1,𝔽8)×D17, …
maximal quotient ofC5×ΓL(1,𝔽8)×Dic51, C2×C5×ΓL(1,𝔽8)×D51

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10710, 36);