Finite Group Databasebeta

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AGL(1,𝔽7)×#700 #72218298

order3360
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22

Aliases

direct productAGL(1,𝔽7)×#700

Splittings

HallD7×#700 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×M16
HallC5×C7 ⋊ C2×C3×M16
HallΓL(1,𝔽8) ⋊ C2×#700
SylowC7 ⋊ C2×C3×#700
SylowC5 ⋊ M16×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, FrattiniC2C2×AGL(1,𝔽5)×AGL(1,𝔽7)
commutator, socleC2×C5×C7C22×C3×C4
FittingC4×C5×C7C2×C3×C4
prime radicalC2×D5×AGL(1,𝔽7)C4
cosocleC7×Dic5C23×C3
hypercenterC4AGL(1,𝔽5)×AGL(1,𝔽7)
hypocenterC5×C7C2×C3×M16
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4

272 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 88 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

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×D5⋊M16
maximal normal subgroup ofAGL(1,𝔽7)×D5⋊M16

GAP

Past the SmallGroups library coverage — no SmallGroups id.