Finite Group Databasebeta

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C5×ΓL(1,𝔽8)×#120 #72218290

order3360
automorphism?
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C24

Aliases

direct productC5×ΓL(1,𝔽8)×#120

Splittings

Fitting, HallC5×C7×#120 ⋊ C3
HallC7×#120 ⋊ C3×C5
HallC5×C7 ⋊ C3×#120
SylowC7 ⋊ C3×C5×#120

Subgroups

subgroupG/H
centerC22×C5C23×ΓL(1,𝔽8)
commutator, cosocleC22×C7C23×C3×C5
FrattiniC22C23×C5×ΓL(1,𝔽8)
FittingC5×C7×#120C3
socleC22×C5×C7C23×C3
hypercenterC5×#120ΓL(1,𝔽8)
hypocenterC7C3×C5×#120
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C5 ◁ C5×#120

520 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 156 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

350 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofD5×ΓL(1,𝔽8)×#120, C5×#120×AGL(1,𝔽7), #120×C5⋊AGL(1,𝔽7)
maximal normal subgroup ofD5×ΓL(1,𝔽8)×#120, C5×#120×AGL(1,𝔽7), #120×C5⋊AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.