Finite Group Databasebeta

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AGL(1,𝔽7)×#63930 #72254208

order12096
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25

Aliases

direct productAGL(1,𝔽7)×#63930

Splittings

HallC32×ΓL(1,𝔽8) ⋊ C22×SD16
SylowC7 ⋊ C2×C3×#63930

Subgroups

subgroupG/H
centerC2AGL(1,𝔽7)×#3811
commutator, cosocleC32×C4×C7C24×C3
FrattiniC4C2×D32×AGL(1,𝔽7)
FittingC32×C7×D4C23×C3
socleC2×C32×C7C22×C3×D4
prime radicalD4×C3⋊D3×AGL(1,𝔽7)C2
hypercenterD4D32×AGL(1,𝔽7)
hypocenterC32×C7C22×C3×SD16
LCSG ▷ C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C4 ◁ D4

3236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id.