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C2×C5×ΓL(1,𝔽8)×D89 #49153153

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

Aliases

direct productC2×C5×ΓL(1,𝔽8)×D89
unambiguous semidirect productC89⋊(C22×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C89)⋊(C2×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D89)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C7×D89)⋊C3

Splittings

Fitting, socleC2×C5×C7×C89 ⋊ C2×C3
derived, HallC7×C89 ⋊ C22×C3×C5
HallC2×C5×C7×D89 ⋊ C3
HallC5×ΓL(1,𝔽8)×C89 ⋊ C22
HallC5×C7×C89 ⋊ C22×C3
HallC2×C7×D89 ⋊ C3×C5
HallΓL(1,𝔽8)×C89 ⋊ C22×C5
HallC5×C89 ⋊ C22×ΓL(1,𝔽8)
SylowC89 ⋊ C22×C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C3×D89
SylowC7 ⋊ C2×C3×C5×D89

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)×D89
commutator, cosocle, hypocenterC7×C89C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C89C2×C3
LCSG ▷ C7×C89
UCSC1 ◁ C2×C5

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 6230

Conjugacy classes & character table

2300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37380, 47);