Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #11721950⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #11721950⋊C2 |
| derived | C22⋊(C22⋊C27) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C27) | C22 |
| Frattini | C22×C9 | C2×S4 |
| Fitting | C9×#120 | D3 |
| socle | C22×C3 | C2×(C22⋊C9)⋊C2 |
| LCS | G ▷ C22⋊(C22⋊C27) | |
| UCS | C1 | |
259 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3^3,2^4|c^3=d,d^3=e,c^a=c^2d^2e^2,d^a=d^2e^2,e^a=e^2,f^a=fg,h^a=hi,f^b=fh,g^b=gi,f^c=g,g^c=fg,h^c=i,i^c=hi54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2197);