Finite Group Databasebeta

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AGL(1,𝔽7)×#911 #72245677

order4032
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productAGL(1,𝔽7)×#911

Splittings

FittingC3×C7×D8 ⋊ C22×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×AGL(1,ℤ/8)
SylowC7 ⋊ C2×C3×#911

Subgroups

subgroupG/H
centerC2D3×D4×AGL(1,𝔽7)
commutator, cosocleC3×C4×C7C24×C3
FrattiniC4C22×D3×AGL(1,𝔽7)
FittingC3×C7×D8C22×C3
socleC2×C3×C7C22×C3×D4
hypercenterD8D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×AGL(1,ℤ/8)
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C4 ◁ D8

1268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×AGL(1,ℤ/8)×AGL(1,𝔽7)
maximal normal subgroup ofD3×AGL(1,ℤ/8)×AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.