Finite Group Databasebeta

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C3×AGL(1,𝔽7)×#213 #72254239

order6048
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC3×AGL(1,𝔽7)×#213

Splittings

FittingC32×C7×D4 ⋊ C22×C3
HallC32×ΓL(1,𝔽8) ⋊ C2×D8
SylowC7 ⋊ C2×C32×#213

Subgroups

subgroupG/H
centerC2×C3#67×AGL(1,𝔽7)
commutator, cosocleC3×C4×C7C23×C32
FrattiniC4C2×C3×D3×AGL(1,𝔽7)
FittingC32×C7×D4C22×C3
socleC2×C32×C7C2×C3×D4
hypercenterC3×D4D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C32×D8
LCSG ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

1344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

252 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#63926, C3×AGL(1,𝔽7)×#932
maximal normal subgroup ofAGL(1,𝔽7)×#63926, C3×AGL(1,𝔽7)×#932

GAP

Past the SmallGroups library coverage — no SmallGroups id.