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C2×C3×ΓL(1,𝔽8)⋊D89 #36375893

order22428
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×ΓL(1,𝔽8)⋊D89
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C3×D89)
unambiguous semidirect product(C2×C89)⋊(C3×AGL(1,𝔽7))
unambiguous semidirect product(C2×D623)⋊C32
unambiguous semidirect product(C2×C3×D623)⋊C3

Splittings

Fitting, socleC2×C3×C7×C89 ⋊ C2×C3
derived, HallC7×C89 ⋊ C22×C32
HallC3×ΓL(1,𝔽8)×C89 ⋊ C22
HallC2×D623 ⋊ C32
SylowC89 ⋊ C2×C3×AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×D89
SylowC7 ⋊ C2×C32×D89

Subgroups

subgroupG/H
center, hypercenterC2×C3ΓL(1,𝔽8)⋊D89
commutator, cosocle, hypocenterC7×C89C22×C32
FrattiniC1G
Fitting, socleC2×C3×C7×C89C2×C3
LCSG ▷ C7×C89
UCSC1 ◁ C2×C3

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3738

Conjugacy classes & character table

1362 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22428, 75);