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C3×C5×ΓL(1,𝔽8)×C53 #30659929

order16695
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C32×C5×C53 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5×ΓL(1,𝔽8)×C53
unambiguous semidirect productC7⋊(C32×C5×C53)
unambiguous semidirect product(C3×C7)⋊(C3×C5×C53)
unambiguous semidirect product(C5×C7)⋊(C32×C53)
unambiguous semidirect product(C3×C5×C7)⋊(C3×C53)
unambiguous semidirect product(C7×C53)⋊(C32×C5)
unambiguous semidirect product(C3×C7×C53)⋊(C3×C5)
unambiguous semidirect product(C5×C7×C53)⋊C32
unambiguous semidirect product(C3×C5×C7×C53)⋊C3

Splittings

Fitting, socleC3×C5×C7×C53 ⋊ C3
derived, SylowC7 ⋊ C32×C5×C53
HallC5×C7×C53 ⋊ C32
HallC7×C53 ⋊ C32×C5
HallC5×C7 ⋊ C32×C53

Subgroups

subgroupG/H
center, hypercenterC3×C5×C53ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C32×C5×C53
FrattiniC1G
Fitting, socleC3×C5×C7×C53C3
LCSG ▷ C7
UCSC1 ◁ C3×C5×C53

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5565

Conjugacy classes & character table

3975 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×C5×AGL(1,𝔽7)×C53, C3×C5×ΓL(1,𝔽8)⋊D53, C3×C5×ΓL(1,𝔽8)×D53, …
maximal normal subgroup ofC3×C5×AGL(1,𝔽7)×C53, C3×C5×ΓL(1,𝔽8)⋊D53, C3×C5×ΓL(1,𝔽8)×D53, …
maximal quotient ofC2×C3×C5×ΓL(1,𝔽8)×C53

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16695, 4);