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#123×AGL(1,𝔽7) #13057390

order1344
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct product#123×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×#123)
unambiguous semidirect product(D7×#123)⋊C3

Splittings

FittingC7×#123 ⋊ C2×C3
HallD7×#123 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#123
SylowC7 ⋊ C2×C3×#123

Subgroups

subgroupG/H
center, FrattiniC22C23×AGL(1,𝔽7)
commutator, socle, cosocleC22×C7C24×C3
FittingC7×#123C2×C3
prime radicalC2×D4×AGL(1,𝔽7)C2
hypercenter#123AGL(1,𝔽7)
hypocenterC7C2×C3×#123
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ #123

684 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#525, AGL(1,𝔽7)×#534, AGL(1,𝔽7)×#546, …
maximal normal subgroup ofAGL(1,𝔽7)×#525, AGL(1,𝔽7)×#534, AGL(1,𝔽7)×#546, …
maximal quotient ofAGL(1,𝔽7)×#385, AGL(1,𝔽7)×#386, AGL(1,𝔽7)×#394

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6679);