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ΓL(1,𝔽8)×C97⋊C3 #20259172

order6111
automorphism?
homologyAbelianization (H1) = C32 · Schur multiplier (H2) = C3

Aliases

direct productΓL(1,𝔽8)×C97⋊C3
unambiguous semidirect product(ΓL(1,𝔽8)×C97)⋊C3
unambiguous semidirect product(C7×C97⋊C3)⋊C3
unambiguous semidirect product#14523271⋊C3
unambiguous semidirect product#14523272⋊C3

Splittings

Fitting, socle, derived, HallC7×C97 ⋊ C32
SylowC97 ⋊ C3×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C97⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC7×C97C32
LCSG ▷ C7×C97
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊(D97⋊C3), AGL(1,𝔽7)×C97⋊C3, ΓL(1,𝔽8)×D97⋊C3, …
maximal normal subgroup ofΓL(1,𝔽8)⋊(D97⋊C3), AGL(1,𝔽7)×C97⋊C3, ΓL(1,𝔽8)×D97⋊C3, …
maximal quotient ofC2×ΓL(1,𝔽8)×C97⋊C3, C22⋊ΓL(1,𝔽8)×C97⋊C3, ΓL(1,𝔽8)×C22⋊(C97⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6111, 7);