Finite Group Databasebeta

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AGL(1,𝔽7)×#4130 #72218428

order6720
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productAGL(1,𝔽7)×#4130

Splittings

FittingC2×C4×C5×C7 ⋊ C2×C3×C4
HallD7×#4130 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×#117
HallC5×C7 ⋊ C2×C3×#117
HallΓL(1,𝔽8) ⋊ C2×#4130
SylowC7 ⋊ C2×C3×#4130
SylowC5 ⋊ #117×AGL(1,𝔽7)

Subgroups

subgroupG/H
centerC4C2×AGL(1,𝔽5)×AGL(1,𝔽7)
commutator, socleC2×C5×C7C23×C3×C4
FrattiniC2C22×AGL(1,𝔽5)×AGL(1,𝔽7)
FittingC2×C4×C5×C7C2×C3×C4
prime radicalC22×D5×AGL(1,𝔽7)C22
cosocleC2×C7×D5C24×C3
hypercenterC2×C4AGL(1,𝔽5)×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×#117
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C4 ◁ C2×C4

1320 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×#65727, AGL(1,𝔽7)×#65767, #41×AGL(1,𝔽5)×AGL(1,𝔽7)
maximal normal subgroup ofAGL(1,𝔽7)×#65727, AGL(1,𝔽7)×#65767, #41×AGL(1,𝔽5)×AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.