Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C33 ⋊ M64 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D3×#157 |
| commutator, socle | C2×C33 | C2×C16 |
| Fitting | C33×C8 | C2×C4 |
| prime radical | C2×C32×D3 | C16 |
| cosocle | C3×C3⋊(C3⋊C16) | C22 |
| hypocenter | C33 | M64 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C8 | |
140 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=c,c^2=d,d^2=e,e^2=f,b^a=bf,h^a=hi,i^a=hi^2,g^b=g^2,h^c=h^2,i^c=i^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20822);