Finite Group Databasebeta

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AGL(1,𝔽7)×#874 #72244268

order4032
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C4

Aliases

direct productAGL(1,𝔽7)×#874

Splittings

FittingC3×C42×C7 ⋊ C22×C3
HallC3×ΓL(1,𝔽8) ⋊ C2×#127
SylowC7 ⋊ C2×C3×#874

Subgroups

subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽7)
commutator, socle, cosocleC22×C3×C7C24×C3
FittingC3×C42×C7C22×C3
hypercenterC42D3×AGL(1,𝔽7)
hypocenterC3×C7C2×C3×#127
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42

2100 conjugacy classes of subgroups; only the 359 normal subgroups are shown.

normal subgroups only

This group has 2100 conjugacy classes of subgroups — too many to draw. Only its 359 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 336

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#127×AGL(1,𝔽7), AGL(1,𝔽7)×#64083, AGL(1,𝔽7)×#4822
maximal normal subgroup ofD3×#127×AGL(1,𝔽7), AGL(1,𝔽7)×#4822
maximal quotient ofAGL(1,𝔽7)×#64083

GAP

Past the SmallGroups library coverage — no SmallGroups id.