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AGL(1,𝔽7)×#687 #72218155

order3360
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productAGL(1,𝔽7)×#687

Splittings

HallD7×#687 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×SD16
HallC5×C7 ⋊ C2×C3×SD16
HallΓL(1,𝔽8) ⋊ C2×#687
SylowC7 ⋊ C2×C3×#687
SylowC5 ⋊ SD16×AGL(1,𝔽7)

Subgroups

subgroupG/H
centerC2C22⋊D5×AGL(1,𝔽7)
commutator, cosocleC4×C5×C7C23×C3
FrattiniC4C2×D5×AGL(1,𝔽7)
FittingC5×C7×D4C22×C3
socleC2×C5×C7C2×C3×D4
prime radicalC5×D4×AGL(1,𝔽7)C2
hypercenterD4D5×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×SD16
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ D4

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

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

Full subgroup data →

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×SD16×AGL(1,𝔽7), AGL(1,𝔽7)×#4057, AGL(1,𝔽7)×#4078
maximal normal subgroup ofD5×SD16×AGL(1,𝔽7), AGL(1,𝔽7)×#4057, AGL(1,𝔽7)×#4078

GAP

Past the SmallGroups library coverage — no SmallGroups id.