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#117×AGL(1,𝔽7) #13057257

order1344
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct product#117×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×#117)
unambiguous semidirect product(D7×#117)⋊C3

Splittings

FittingC7×#117 ⋊ C2×C3
HallD7×#117 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#117
SylowC7 ⋊ C2×C3×#117

Subgroups

subgroupG/H
centerC2×C4C22×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC22C23×AGL(1,𝔽7)
FittingC7×#117C2×C3
socle, cosocleC22×C7C24×C3
prime radicalC23×AGL(1,𝔽7)C22
hypercenter#117AGL(1,𝔽7)
hypocenterC7C2×C3×#117
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #117

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#4130, AGL(1,𝔽7)×#428, AGL(1,𝔽7)×#418, …
maximal normal subgroup ofAGL(1,𝔽7)×#428, AGL(1,𝔽7)×#418, C4×#41×AGL(1,𝔽7), …
maximal quotient ofAGL(1,𝔽7)×#4130, C4×C22⋊C4×AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6546);