Finite Group Databasebeta

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ΓL(1,𝔽8)×#4099 #72218297

order3360
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C24

Aliases

direct productΓL(1,𝔽8)×#4099

Splittings

FittingC24×C5×C7 ⋊ C2×C3
HallC7×#4099 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ #120
HallC5×C7 ⋊ C3×#120
SylowC7 ⋊ C3×#4099
SylowC5 ⋊ ΓL(1,𝔽8)×#120

Subgroups

subgroupG/H
center, FrattiniC22C22×D5×ΓL(1,𝔽8)
commutator, socle, cosocleC22×C5×C7C23×C3
FittingC24×C5×C7C2×C3
hypercenterC24D5×ΓL(1,𝔽8)
hypocenterC5×C7C3×#120
LCSG ▷ C22×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C24

520 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 123 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

230 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×ΓL(1,𝔽8)×#120, AGL(1,𝔽7)×#4099, #72218525
maximal normal subgroup ofD5×ΓL(1,𝔽8)×#120, AGL(1,𝔽7)×#4099, #72218525

GAP

Past the SmallGroups library coverage — no SmallGroups id.