Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #458 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11722947 |
| commutator, Fitting, cosocle | C23×C33 | C23 |
| Frattini, hypercenter | C23 | D33 |
| socle | C2×C33 | #120 |
| prime radical | C3×#64330 | C2 |
| hypocenter | C33 | #458 |
| LCS | G ▷ C23×C33 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C23 | |
856 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 856 conjugacy classes of subgroups — too many to draw. Only its 98 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|c^2=ef,c^a=ce,d^a=df,i^a=i^2,c^b=cd,e^b=ef,g^b=g^2,d^c=df,h^c=h^296 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23056);