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AGL(1,ℤ/8)×AGL(1,𝔽7) #13057506

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

Aliases

direct productAGL(1,ℤ/8)×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×AGL(1,ℤ/8))
unambiguous semidirect product(D7×AGL(1,ℤ/8))⋊C3

Splittings

FittingC7×AGL(1,ℤ/8) ⋊ C2×C3
HallD7×AGL(1,ℤ/8) ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×AGL(1,ℤ/8)
SylowC7 ⋊ C2×C3×AGL(1,ℤ/8)

Subgroups

subgroupG/H
centerC2C2×D4×AGL(1,𝔽7)
commutator, cosocleC4×C7C24×C3
FrattiniC4C23×AGL(1,𝔽7)
FittingC7×AGL(1,ℤ/8)C2×C3
socleC2×C7C22×C3×D4
hypercenterAGL(1,ℤ/8)AGL(1,𝔽7)
hypocenterC7C2×C3×AGL(1,ℤ/8)
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 56

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#453, #72214550, C5×AGL(1,ℤ/8)×AGL(1,𝔽7), …
maximal normal subgroup ofAGL(1,𝔽7)×#453, C5×AGL(1,ℤ/8)×AGL(1,𝔽7), C2×AGL(1,ℤ/8)×AGL(1,𝔽7), …
maximal quotient of#72214550, C5×AGL(1,ℤ/8)×AGL(1,𝔽7), AGL(1,ℤ/8)×C5⋊AGL(1,𝔽7), …
automorphism group ofD56, Dic28, D56⋊C3, …
holomorph ofC7×C8

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6795);