Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊(C3⋊#857) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C33 |
| unambiguous semidirect product | C3⋊(C3⋊#4623) |
| derived | C32×C42⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊#857) |
| commutator, hypocenter | C32×C42⋊C3 | C4 |
| Frattini | C23 | C3⋊(C3⋊S4) |
| Fitting | C2×C32×C42 | D3 |
| socle | C23×C32 | S4 |
| prime radical, cosocle | C2×C32×C42⋊C3 | C2 |
| LCS | G ▷ C32×C42⋊C3 | |
| UCS | C1 ◁ C2 | |
432 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 81 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|a^2=b,f^2=h,g^2=i,c^a=c^2,d^a=d^2,e^a=e^2,f^a=fgi,g^a=gi,h^a=hi,f^c=gh,g^c=fg,h^c=i,i^c=hi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23080);