Finite Group Databasebeta

← Prev (#13153808) Next (#13153810) →

C2×#11721733 #13153809

order1728
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4

Aliases

direct productC2×#11721733

Splittings

SylowH(3,𝔽3) ⋊ C2×#117

Subgroups

subgroupG/H
center, socleC23×C3C22×C3⋊D3
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C23×C3⋊D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC24×H(3,𝔽3)C22
cosocleC22×H(3,𝔽3)C24
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterH(3,𝔽3)C2×#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

1815 conjugacy classes of subgroups; only the 357 normal subgroups are shown.

normal subgroups only

This group has 1815 conjugacy classes of subgroups — too many to draw. Only its 357 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

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 19872);