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C5×ΓL(1,𝔽8)⋊(D43⋊C3) #41029129

order27090
automorphism?
homologyAbelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×ΓL(1,𝔽8)⋊(D43⋊C3)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D43⋊C3)
unambiguous semidirect product(C43⋊C3)⋊(C5×AGL(1,𝔽7))
unambiguous semidirect product((C43⋊C3)⋊D7)⋊(C3×C5)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D43)⋊(C3×C5)
unambiguous semidirect product#14268593⋊(C3×C5)
unambiguous semidirect product#14268594⋊(C3×C5)
unambiguous semidirect product(C5×(C43⋊C3)⋊D7)⋊C3
unambiguous semidirect product(C5×ΓL(1,𝔽8)⋊D43)⋊C3
unambiguous semidirect product(C5×#14268594)⋊C3
unambiguous semidirect product(C5×#14268593)⋊C3

Splittings

Fitting, socle, HallC5×C7×C43 ⋊ C2×C32
derived, HallC7×C43 ⋊ C2×C32×C5
HallC5×ΓL(1,𝔽8)×C43⋊C3 ⋊ C2
HallC5×D301 ⋊ C32
HallΓL(1,𝔽8)×C43⋊C3 ⋊ C2×C5
HallD301 ⋊ C32×C5
HallC5×C43 ⋊ C3×AGL(1,𝔽7)
SylowC43 ⋊ C3×C5×AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D43⋊C3
SylowC7 ⋊ C3×C5×D43⋊C3

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)⋊(D43⋊C3)
commutator, cosocle, hypocenterC7×C43C2×C32×C5
FrattiniC1G
Fitting, socleC5×C7×C43C2×C32
LCSG ▷ C7×C43
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1505

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27090, 70);