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AGL(1,𝔽7)×#496 #72259374

order2688
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4

Aliases

direct productAGL(1,𝔽7)×#496

Splittings

FittingC7×#496 ⋊ C2×C3
HallD7×#496 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×#496
SylowC7 ⋊ C2×C3×#496

Subgroups

subgroupG/H
centerC22C2×D4×AGL(1,𝔽7)
commutator, cosocleC2×C4×C7C24×C3
FrattiniC2×C4C23×AGL(1,𝔽7)
FittingC7×#496C2×C3
socleC22×C7C22×C3×D4
hypercenter#496AGL(1,𝔽7)
hypocenterC7C2×C3×#496
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #496

1336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#3151
maximal normal subgroup ofAGL(1,𝔽7)×#3151
maximal quotient ofAGL(1,𝔽7)×#1715, AGL(1,𝔽7)×#1721

GAP

Past the SmallGroups library coverage — no SmallGroups id.