Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C3 |
| unambiguous semidirect product | #10596188⋊C3 |
| Fitting | H(3,𝔽3)×#127 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×(C3⋊D3)⋊C3 |
| commutator, cosocle | C22×C32 | C24×C3 |
| Frattini, socle | C22×C3 | C23×C3×D3 |
| Fitting | H(3,𝔽3)×#127 | C2 |
| hypercenter | #127 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×D42 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #127 | |
1606 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 1606 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^4,3,2^2,3^2|b^2=g,b^a=bg,h^a=h^2,i^a=i^2,c^b=cg,d^b=dg,d^c=df,h^e=hi131 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19339);