Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of (#64188⋊C2)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (#64188⋊C2)⋊C3 |
| unambiguous semidirect product | #11724050⋊C2 |
| derived | C22⋊(C22⋊C9) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C22×C3 |
| Frattini, socle | C22×C3 | C2×C3×S4 |
| Fitting | C32×#120 | D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 | |
385 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 385 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3^3,2^4|d^3=e,d^a=d^2e^2,e^a=e^2,f^a=fg,h^a=hi,f^b=fh,g^b=gi,d^c=de,f^d=g,g^d=fg,h^d=i,i^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12893);