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ΓL(1,𝔽8)×D445 #34035086

order18690
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D445
unambiguous semidirect productC5⋊(ΓL(1,𝔽8)×D89)
unambiguous semidirect productC89⋊(D5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D445)⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C89 ⋊ C2×C3
HallC5×ΓL(1,𝔽8)×C89 ⋊ C2
HallC7×D445 ⋊ C3
HallΓL(1,𝔽8)×C89 ⋊ D5
HallC7×C89 ⋊ C3×D5
HallC5×C89 ⋊ C2×ΓL(1,𝔽8)
HallC5×ΓL(1,𝔽8) ⋊ D89
SylowC89 ⋊ D5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×D89
SylowC7 ⋊ C3×D445
SylowC5 ⋊ ΓL(1,𝔽8)×D89

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C7×C89C2×C3
LCSG ▷ C5×C7×C89
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3115

Conjugacy classes & character table

1120 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊#13184068, ΓL(1,𝔽8)⋊#13184067, ΓL(1,𝔽8)×#13184068, …
maximal normal subgroup ofΓL(1,𝔽8)⋊#13184068, ΓL(1,𝔽8)⋊#13184067, ΓL(1,𝔽8)×#13184068, …
maximal quotient ofΓL(1,𝔽8)×Dic445, C2×ΓL(1,𝔽8)×D445

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18690, 22);