Finite Group Databasebeta

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C22×#88538 #13154406

order1728
isoclinismisoclinic class of #6229
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC22×#88538

Splittings

SylowH(3,𝔽3) ⋊ C22×M16

Subgroups

subgroupG/H
centerC22×C3×C4C2×C3⋊D3
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC3×C4C23×C3⋊D3
FittingC23×C4×H(3,𝔽3)C2
socleC23×C3C2×C3⋊Dic3
prime radicalC24×H(3,𝔽3)C4
cosocleC4×H(3,𝔽3)C24
hypercenterC23×C3×C4C3⋊D3
hypocenterH(3,𝔽3)C22×M16
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22×C3×C4 ◁ C23×C3×C4

1639 conjugacy classes of subgroups; only the 597 normal subgroups are shown.

normal subgroups only

This group has 1639 conjugacy classes of subgroups — too many to draw. Only its 597 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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, 20469);