Finite Group Databasebeta

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C2×#11720573 #13144804

order1728
automorphism?
homologyAbelianization (H1) = C22×C8 · Schur multiplier (H2) = C24

Aliases

direct productC2×#11720573

Splittings

SylowH(3,𝔽3) ⋊ C2×C22⋊C8

Subgroups

subgroupG/H
centerC22×C3×C4C2×C3⋊D3
commutatorC2×H(3,𝔽3)C22×C8
FrattiniC2×C3×C4C22×C3⋊D3
FittingC23×C4×H(3,𝔽3)C2
socleC23×C3C2×C3⋊Dic3
prime radicalC24×H(3,𝔽3)C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC23×C3×C4C3⋊D3
hypocenterH(3,𝔽3)C2×C22⋊C8
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3×C4 ◁ C23×C3×C4

1111 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 288

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10867);