Finite Group Databasebeta

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C22×D3×H(3,𝔽3) #10620106

order648
automorphism?
homologyAbelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32

Aliases

direct productC22×D3×H(3,𝔽3)
unambiguous semidirect productC3⋊(C23×H(3,𝔽3))
unambiguous semidirect product(C2×C3)⋊(C22×H(3,𝔽3))

Splittings

FittingC22×C3×H(3,𝔽3) ⋊ C2
SylowC3×H(3,𝔽3) ⋊ C23

Subgroups

subgroupG/H
centerC22×C3C32×D3
commutator, cosocleC32C23×C32
FrattiniC3C22×C32×D3
FittingC22×C3×H(3,𝔽3)C2
socleC22×C32C2×C32
hypercenterC22×H(3,𝔽3)D3
hypocenterC3C23×H(3,𝔽3)
LCSG ▷ C32 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×H(3,𝔽3)

437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D3)⋊((C3⋊Dic3)⋊C3), (C2×D3)⋊#1084, H(3,𝔽3)×(C2×D3)⋊C4, …
maximal normal subgroup of(C2×D3)⋊((C3⋊Dic3)⋊C3), (C2×D3)⋊#1084, H(3,𝔽3)×(C2×D3)⋊C4, …
maximal quotient ofC2×Dic6×H(3,𝔽3), C2×C4×D3×H(3,𝔽3), C2×D12×H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(648, 625);