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C5×ΓL(1,𝔽8)⋊(D37⋊C3) #37552036

order23310
automorphism?
homologyAbelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×ΓL(1,𝔽8)⋊(D37⋊C3)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D37⋊C3)
unambiguous semidirect product(C37⋊C3)⋊(C5×AGL(1,𝔽7))
unambiguous semidirect product((C37⋊C3)⋊D7)⋊(C3×C5)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D37)⋊(C3×C5)
unambiguous semidirect product#13095537⋊(C3×C5)
unambiguous semidirect product#13095538⋊(C3×C5)
unambiguous semidirect product(C5×(C37⋊C3)⋊D7)⋊C3
unambiguous semidirect product(C5×ΓL(1,𝔽8)⋊D37)⋊C3
unambiguous semidirect product(C5×#13095537)⋊C3
unambiguous semidirect product(C5×#13095538)⋊C3

Splittings

Fitting, socle, HallC5×C7×C37 ⋊ C2×C32
derived, HallC7×C37 ⋊ C2×C32×C5
HallC5×ΓL(1,𝔽8)×C37⋊C3 ⋊ C2
HallC5×D259 ⋊ C32
HallΓL(1,𝔽8)×C37⋊C3 ⋊ C2×C5
HallD259 ⋊ C32×C5
HallC5×C37 ⋊ C3×AGL(1,𝔽7)
SylowC37 ⋊ C3×C5×AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D37⋊C3
SylowC7 ⋊ C3×C5×D37⋊C3

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)⋊(D37⋊C3)
commutator, cosocle, hypocenterC7×C37C2×C32×C5
FrattiniC1G
Fitting, socleC5×C7×C37C2×C32
LCSG ▷ C7×C37
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1295

Conjugacy classes & character table

255 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 55);