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ΓL(1,𝔽8)×C5⋊(D37⋊C3) #37552055

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

Aliases

direct productΓL(1,𝔽8)×C5⋊(D37⋊C3)
unambiguous semidirect productC5⋊(ΓL(1,𝔽8)×D37⋊C3)
unambiguous semidirect product(C37⋊C3)⋊(D5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×C5⋊(D37⋊C3))⋊C3
unambiguous semidirect product(ΓL(1,𝔽8)×D185)⋊C3
unambiguous semidirect product(C5⋊#13095531)⋊C3
unambiguous semidirect product(C5⋊#13095532)⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C37 ⋊ C2×C32
HallC5×ΓL(1,𝔽8)×C37⋊C3 ⋊ C2
HallC7×D185 ⋊ C32
HallΓL(1,𝔽8)×C37⋊C3 ⋊ D5
HallD185 ⋊ C3×ΓL(1,𝔽8)
HallC7×C37 ⋊ C32×D5
HallC5×C37 ⋊ C2×C3×ΓL(1,𝔽8)
SylowC37 ⋊ C3×D5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×D37⋊C3
SylowC7 ⋊ C3×C5⋊(D37⋊C3)
SylowC5 ⋊ ΓL(1,𝔽8)×D37⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C37C2×C32
LCSG ▷ C5×C7×C37
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1295

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 74);