Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C33 ⋊ #393 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C2×#1113 |
| commutator, socle, cosocle | C23×C33 | C23 |
| Fitting, prime radical | C24×C33 | C22 |
| hypercenter | C24 | #1113 |
| hypocenter | C33 | #393 |
| LCS | G ▷ C23×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ C24 | |
1338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 1338 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=f,b^a=bf,c^a=ce,g^a=g^2,h^a=h^2,c^b=cd,g^b=g^2,i^b=i^2162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 28126);