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C5⋊(ΓL(1,𝔽8)⋊(D37⋊C3)) #37552049

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

Aliases

unambiguous semidirect 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(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, derived, HallC5×C7×C37 ⋊ C2×C32
HallC5×ΓL(1,𝔽8)×C37⋊C3 ⋊ C2
HallD1295 ⋊ C32
HallΓL(1,𝔽8)×C37⋊C3 ⋊ D5
HallC7×C37 ⋊ C32×D5
HallC5×C37 ⋊ C3×AGL(1,𝔽7)
SylowC37 ⋊ C3×C5⋊AGL(1,𝔽7)
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

Polycyclic presentation:

2,3^2,5,7,37|d^a=d^4,e^a=e^6,f^a=f^36,f^b=f^26,e^c=e^2

Conjugacy classes & character table

201 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 68);