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#135×AGL(1,𝔽7) #13057501

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

Aliases

direct product#135×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×#135)
unambiguous semidirect product(D7×#135)⋊C3

Splittings

FittingC7×#135 ⋊ C2×C3
HallD7×#135 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#135
SylowC7 ⋊ C2×C3×#135

Subgroups

subgroupG/H
centerC4D4×AGL(1,𝔽7)
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×AGL(1,𝔽7)
FittingC7×#135C2×C3
socleC2×C7C22×C3×D4
hypercenter#135AGL(1,𝔽7)
hypocenterC7C2×C3×#135
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

532 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)×#575, AGL(1,𝔽7)×#509, AGL(1,𝔽7)×#360, …
maximal normal subgroup ofAGL(1,𝔽7)×#575, AGL(1,𝔽7)×#509, AGL(1,𝔽7)×#360, …
maximal quotient ofC4×D8×AGL(1,𝔽7), C4×SD16×AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6790);