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C2×ΓL(1,𝔽8)×C73⋊C3 #22604116

order9198
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC2×ΓL(1,𝔽8)×C73⋊C3
unambiguous semidirect product(C2×C7×C73⋊C3)⋊C3
unambiguous semidirect product(C2×ΓL(1,𝔽8)×C73)⋊C3
unambiguous semidirect product(C2×#13095275)⋊C3
unambiguous semidirect product(C2×#13095276)⋊C3

Splittings

Fitting, socle, HallC2×C7×C73 ⋊ C32
derived, HallC7×C73 ⋊ C2×C32
HallC2×C73 ⋊ C3×ΓL(1,𝔽8)
SylowC73 ⋊ C2×C3×ΓL(1,𝔽8)
HallC2×C7 ⋊ C3×C73⋊C3
SylowC7 ⋊ C2×C3×C73⋊C3

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×C73⋊C3
commutator, hypocenterC7×C73C2×C32
FrattiniC1G
Fitting, socleC2×C7×C73C32
LCSG ▷ C7×C73
UCSC1 ◁ C2

48 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

270 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊(Dic73⋊C3), ΓL(1,𝔽8)⋊C4×C73⋊C3, ΓL(1,𝔽8)×Dic73⋊C3, …
maximal normal subgroup ofΓL(1,𝔽8)⋊(Dic73⋊C3), ΓL(1,𝔽8)⋊C4×C73⋊C3, ΓL(1,𝔽8)×Dic73⋊C3, …
maximal quotient ofC4×ΓL(1,𝔽8)×C73⋊C3, C22×ΓL(1,𝔽8)×C73⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9198, 41);