Finite Group Databasebeta

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C3×#89804⋊C3 #72246159

order4032
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3

Aliases

direct productC3×#89804⋊C3

Splittings

FittingC3×C7×#120 ⋊ C2×C3
Hall#89804 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ B(4,𝔽2)
SylowC7 ⋊ C32×B(4,𝔽2)

Subgroups

subgroupG/H
centerC2×C3#6500⋊C3
commutator, cosocleC23×C7C23×C32
FrattiniC23C22×C3×AGL(1,𝔽7)
FittingC3×C7×#120C2×C3
socleC2×C3×C7C3×#120
hypercenterC3×#120AGL(1,𝔽7)
hypocenterC7C32×B(4,𝔽2)
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#120

1188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

186 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×B(4,𝔽2), #72246319, D3×#89804⋊C3, …
maximal normal subgroup ofC3×AGL(1,𝔽7)×B(4,𝔽2), D3×#89804⋊C3, #72269452
maximal quotient of#72246319

GAP

Past the SmallGroups library coverage — no SmallGroups id.