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#137×AGL(1,𝔽7) #13057510

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

Aliases

direct product#137×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×#137)
unambiguous semidirect product(D7×#137)⋊C3

Splittings

FittingC7×#137 ⋊ C2×C3
HallD7×#137 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#137
SylowC7 ⋊ C2×C3×#137

Subgroups

subgroupG/H
centerC2C2×D4×AGL(1,𝔽7)
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×AGL(1,𝔽7)
FittingC7×#137C2×C3
socleC2×C7C22×C3×D4
prime radical#41×AGL(1,𝔽7)C2
hypercenter#137AGL(1,𝔽7)
hypocenterC7C2×C3×#137
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#575, AGL(1,𝔽7)×#577, AGL(1,𝔽7)×#455
maximal normal subgroup ofAGL(1,𝔽7)×#575, AGL(1,𝔽7)×#577, AGL(1,𝔽7)×#455
maximal quotient ofAGL(1,𝔽7)×#428, AGL(1,𝔽7)×#440, AGL(1,𝔽7)×#417

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6799);