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ΓL(1,𝔽8)⋊D89 #16836653

order3738
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊D89
unambiguous semidirect productC89⋊AGL(1,𝔽7)
unambiguous semidirect productD623⋊C3

Splittings

Fitting, socle, derived, HallC7×C89 ⋊ C2×C3
HallΓL(1,𝔽8)×C89 ⋊ C2
HallD623 ⋊ C3
SylowC89 ⋊ AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ D89
SylowC7 ⋊ C3×D89

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C89C2×C3
LCSG ▷ C7×C89
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 96
transitive degreeμt(G) = 623

Polycyclic presentation:

2,3,7,89|c^a=c^6,d^a=d^88,c^b=c^2

Conjugacy classes & character table

227 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)⋊D89, C3⋊(ΓL(1,𝔽8)⋊D89), C5×ΓL(1,𝔽8)⋊D89, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)⋊D89, C5×ΓL(1,𝔽8)⋊D89, C7×ΓL(1,𝔽8)⋊D89
maximal quotient of(C7⋊C9)⋊D89, C3×ΓL(1,𝔽8)⋊D89, C3⋊(ΓL(1,𝔽8)⋊D89), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3738, 12);