Finite Group Databasebeta

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C22×#88671 #13164812

order1728
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC22×#88671

Splittings

SylowH(3,𝔽3) ⋊ C22×#41

Subgroups

subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C25
FrattiniC2×C3C23×D32
FittingC24×H(3,𝔽3)C22
socleC23×C3C2×D32
hypercenterC24((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C24

3351 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 288

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 30875);