Finite Group Databasebeta

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AGL(1,𝔽7)×#834 #72254374

order4032
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productAGL(1,𝔽7)×#834

Splittings

HallC3×ΓL(1,𝔽8) ⋊ C2×#99
SylowC7 ⋊ C2×C3×#834

Subgroups

subgroupG/H
centerC2AGL(1,𝔽7)×#217
commutatorC22×C3×C7C22×C3×C4
FrattiniC23C2×D3×AGL(1,𝔽7)
FittingC2×C3×C7×D4C22×C3
socleC2×C3×C7C2×C3×C22⋊C4
prime radicalC2×C3×D4×AGL(1,𝔽7)C2
cosocleC23×C3×C7C23×C3
hypercenterC2×D4D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×#99
LCSG ▷ C22×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

1076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#99×AGL(1,𝔽7), AGL(1,𝔽7)×#5244, AGL(1,𝔽7)×#5032
maximal normal subgroup ofD3×#99×AGL(1,𝔽7), AGL(1,𝔽7)×#5244, AGL(1,𝔽7)×#5032

GAP

Past the SmallGroups library coverage — no SmallGroups id.