Finite Group Databasebeta

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AGL(1,𝔽7)×#731 #72249133

order3402
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C33

Aliases

direct productAGL(1,𝔽7)×#731

Splittings

FittingC7×#731 ⋊ C2×C3
HallΓL(1,𝔽8)×#731 ⋊ C2
HallD7 ⋊ C3×#731
SylowC7 ⋊ C2×C3×#731

Subgroups

subgroupG/H
centerC3H(3,𝔽3)×AGL(1,𝔽7)
commutator, cosocleC32×C7C2×C33
FrattiniC32C32×AGL(1,𝔽7)
FittingC7×#731C2×C3
socleC3×C7C2×C3×H(3,𝔽3)
prime radicalH(3,𝔽3)×AGL(1,𝔽7)C3
hypercenter#731AGL(1,𝔽7)
hypocenterC7C2×C3×#731
LCSG ▷ C32×C7 ▷ C3×C7 ▷ C7
UCSC1 ◁ C3 ◁ C32 ◁ #731

288 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 189

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#6910, AGL(1,𝔽7)×#4184
maximal normal subgroup ofAGL(1,𝔽7)×#6910, AGL(1,𝔽7)×#4184

GAP

Past the SmallGroups library coverage — no SmallGroups id.