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C5×AGL(1,𝔽7)×D89 #49153133

order37380
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×AGL(1,𝔽7)×D89
unambiguous semidirect productD7⋊(C3×C5×D89)
unambiguous semidirect product(D7×D89)⋊(C3×C5)
unambiguous semidirect product(ΓL(1,𝔽8)×D89)⋊(C2×C5)
unambiguous semidirect product(AGL(1,𝔽7)×C89)⋊(C2×C5)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D89)⋊(C2×C5)
unambiguous semidirect product(C5×D7×D89)⋊C3

Splittings

Fitting, socle, HallC5×C7×C89 ⋊ C22×C3
derived, HallC7×C89 ⋊ C22×C3×C5
HallC5×D7×D89 ⋊ C3
HallC5×ΓL(1,𝔽8)×C89 ⋊ C22
HallD7×D89 ⋊ C3×C5
HallΓL(1,𝔽8)×C89 ⋊ C22×C5
HallC5×C89 ⋊ C2×AGL(1,𝔽7)
HallC5×ΓL(1,𝔽8) ⋊ C2×D89
SylowC89 ⋊ C2×C5×AGL(1,𝔽7)
HallC5×C7 ⋊ C2×C3×D89
HallΓL(1,𝔽8) ⋊ C2×C5×D89
SylowC7 ⋊ C2×C3×C5×D89

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽7)×D89
commutator, cosocle, hypocenterC7×C89C22×C3×C5
FrattiniC1G
Fitting, socleC5×C7×C89C22×C3
LCSG ▷ C7×C89
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3115

Conjugacy classes & character table

1610 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37380, 27);