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ΓL(1,𝔽8)×D267 #26014253

order11214
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D267
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×D89)
unambiguous semidirect productC89⋊(D3×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D267)⋊C3

Splittings

Fitting, socle, derivedC3×C7×C89 ⋊ C2×C3
HallC3×ΓL(1,𝔽8)×C89 ⋊ C2
HallC7×C89 ⋊ C3×D3
SylowC89 ⋊ D3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ D89
SylowC7 ⋊ C3×D267

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C89C2×C3
LCSG ▷ C3×C7×C89
UCSC1

36 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

675 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D267, (C3⋊AGL(1,𝔽7))⋊D89, D3×ΓL(1,𝔽8)×D89, …
maximal normal subgroup ofAGL(1,𝔽7)×D267, (C3⋊AGL(1,𝔽7))⋊D89, D3×ΓL(1,𝔽8)×D89, …
maximal quotient ofΓL(1,𝔽8)×Dic267, C2×ΓL(1,𝔽8)×D267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(11214, 20);