Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10593847⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | #10593847⋊C3 |
| Fitting, derived | C24×C32 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C24×C32 | C3×C4 |
| Frattini | C22 | A4×#157 |
| socle | C22×C32 | C4×A4 |
| prime radical | C3⋊D3×C22⋊A4 | C2 |
| cosocle | C24×C3⋊D3 | C2×C3 |
| LCS | G ▷ C24×C32 | |
| UCS | C1 | |
536 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 536 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2,3,2^5,3^2|a^2=c,d^a=df,e^a=efg,h^a=hi,i^a=hi^2,d^b=de,e^b=d,f^b=g,g^b=fg,h^c=h^2,i^c=i^236 conjugacy classes — view the full character table.
| maximal subgroup of | #72241536, #72241537, #72241539 |
| maximal normal subgroup of | #72241536, #72241537, #72241539 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31909);