Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #338 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #88813 |
| commutator | C33×C4 | C42 |
| Frattini, hypercenter | C22×C4 | #1113 |
| Fitting, cosocle | C22×C33×C4 | C22 |
| socle | C2×C33 | #95 |
| prime radical | C23×C33 | Q8 |
| hypocenter | C33 | #338 |
| LCS | G ▷ C33×C4 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ C22×C4 | |
602 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 602 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|a^2=e,b^2=c,c^2=f,d^2=f,e^2=f,b^a=bd,c^a=cf,g^a=g^2,i^a=i^2,e^b=ef,g^b=g^2,h^b=h^2162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11796);