Finite Group Databasebeta

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C3×D12×AGL(1,𝔽7) #72246941

order3024
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3

Aliases

direct productC3×D12×AGL(1,𝔽7)

Splittings

FittingC32×C4×C7 ⋊ C22×C3
HallC32×ΓL(1,𝔽8) ⋊ C2×D4
SylowC7 ⋊ C2×C32×D12

Subgroups

subgroupG/H
centerC2×C3C2×D3×AGL(1,𝔽7)
commutator, cosocleC2×C3×C7C23×C32
FrattiniC2C2×C3×D3×AGL(1,𝔽7)
FittingC32×C4×C7C22×C3
socleC2×C32×C7C23×C3
hypercenterC3×C4D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C32×D4
LCSG ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2×C3 ◁ C3×C4

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

189 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#3802, C3×D3×D4×AGL(1,𝔽7), D3×D12×AGL(1,𝔽7), …
maximal normal subgroup ofAGL(1,𝔽7)×#3802, C3×D3×D4×AGL(1,𝔽7), D3×D12×AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.