Finite Group Databasebeta

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C2×C3×D4×AGL(1,𝔽7) #72244107

order2016
automorphism?
homologyAbelianization (H1) = C24×C32 · Schur multiplier (H2) = C26×C3

Aliases

direct productC2×C3×D4×AGL(1,𝔽7)

Splittings

FittingC2×C3×C7×D4 ⋊ C2×C3
HallC2×D4×D7 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C22×D4
SylowC7 ⋊ C22×C32×D4

Subgroups

subgroupG/H
centerC22×C3C22×AGL(1,𝔽7)
commutator, cosocleC2×C7C24×C32
FrattiniC2C23×C3×AGL(1,𝔽7)
FittingC2×C3×C7×D4C2×C3
socleC22×C3×C7C23×C3
hypercenterC2×C3×D4AGL(1,𝔽7)
hypocenterC7C22×C32×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

1416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D3×D4×AGL(1,𝔽7), C3×#120×AGL(1,𝔽7), AGL(1,𝔽7)×#937, …
maximal normal subgroup ofC2×D3×D4×AGL(1,𝔽7), C3×#120×AGL(1,𝔽7), AGL(1,𝔽7)×#937, …
maximal quotient ofC3×#120×AGL(1,𝔽7), C3×#127×AGL(1,𝔽7), C3×AGL(1,ℤ/8)×AGL(1,𝔽7), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.