Finite Group Databasebeta

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C5×ΓL(1,𝔽8)×C22⋊D5 #72243164

order4200
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×ΓL(1,𝔽8)×C22⋊D5

Splittings

FittingC22×C52×C7 ⋊ C2×C3
HallC5×C7×C22⋊D5 ⋊ C3
HallC52×ΓL(1,𝔽8) ⋊ D4
HallC52×C7 ⋊ C3×D4
SylowC52 ⋊ D4×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C5×C22⋊D5

Subgroups

subgroupG/H
centerC2×C5C2×D5×ΓL(1,𝔽8)
commutator, cosocleC2×C5×C7C22×C3×C5
FrattiniC2C2×C5×D5×ΓL(1,𝔽8)
FittingC22×C52×C7C2×C3
socleC2×C52×C7C22×C3
hypercenterC22×C5D5×ΓL(1,𝔽8)
hypocenterC5×C7C3×C5×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2×C5 ◁ C22×C5

168 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Conjugacy classes & character table

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

Relationships

maximal subgroup ofΓL(1,𝔽8)×#87886, C5×C22⋊D5×AGL(1,𝔽7), #72243211
maximal normal subgroup ofΓL(1,𝔽8)×#87886, C5×C22⋊D5×AGL(1,𝔽7), #72243211

GAP

Past the SmallGroups library coverage — no SmallGroups id.