Finite Group Databasebeta

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C3×AGL(1,𝔽7)×#235 #72247143

order6048
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C25×C3

Aliases

direct productC3×AGL(1,𝔽7)×#235

Splittings

FittingC2×C32×C4×C7 ⋊ C22×C3
HallC32×ΓL(1,𝔽8) ⋊ C2×#41
SylowC7 ⋊ C2×C32×#235

Subgroups

subgroupG/H
centerC3×C4C2×D3×AGL(1,𝔽7)
commutator, cosocleC2×C3×C7C24×C32
FrattiniC2C22×C3×D3×AGL(1,𝔽7)
FittingC2×C32×C4×C7C22×C3
socleC2×C32×C7C24×C3
hypercenterC2×C3×C4D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C32×#41
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

2400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

378 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#64306, C3×AGL(1,𝔽7)×#932
maximal normal subgroup ofAGL(1,𝔽7)×#64306, C3×AGL(1,𝔽7)×#932

GAP

Past the SmallGroups library coverage — no SmallGroups id.