Finite Group Databasebeta

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C3×#6500⋊C3 #72244132

order2016
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productC3×#6500⋊C3

Splittings

FittingC2×C3×C7×D4 ⋊ C2×C3
Hall#6500 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ #120
SylowC7 ⋊ C32×#120

Subgroups

subgroupG/H
centerC22×C3C22×AGL(1,𝔽7)
commutator, cosocleC22×C7C23×C32
FrattiniC22C22×C3×AGL(1,𝔽7)
FittingC2×C3×C7×D4C2×C3
socleC22×C3×C7C23×C3
hypercenterC2×C3×D4AGL(1,𝔽7)
hypocenterC7C32×#120
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×#120×AGL(1,𝔽7), D3×#6500⋊C3, #72244193, …
maximal normal subgroup ofC3×#120×AGL(1,𝔽7), D3×#6500⋊C3, #72244193, …
maximal quotient ofC3×#89849⋊C3, C3×#89970⋊C3, #72246099, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.