Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #13147121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×#126 ⋊ C2×C3 |
| derived | C42×C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C9×#126 | C2×C3 |
| LCS | G ▷ C42×C9 | |
| UCS | C1 | |
207 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3,2^2,3,2^2,3|d^2=g,e^2=gh,f^3=i,d^a=dg,e^a=egh,f^a=f^2i^2,i^a=i^2,d^b=dgh,e^b=eh,d^c=de,e^c=dgh,f^c=fi^2,g^c=h,h^c=gh41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13184);