Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C33 |
No aliases for this group.
| Fitting | C22×C33×D4 ⋊ C2 |
| Sylow | C33 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C23×C33 | C23 |
| Fitting | C22×C33×D4 | C2 |
| hypercenter | C22×D4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #392 |
| LCS | G ▷ C23×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
4508 conjugacy classes of subgroups; only the 745 normal subgroups are shown.
This group has 4508 conjugacy classes of subgroups — too many to draw. Only its 745 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^6,3^3|b^a=bf,c^a=ce,g^a=g^2,h^a=h^2,i^a=i^2,c^b=cd282 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 29926);