Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10588306⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #10588306⋊C3 |
| derived | C42⋊C9 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C42⋊C9 | C3×C4 |
| Frattini | C23×C3 | C3×S4 |
| Fitting, prime radical | C32×#127 | D3 |
| socle | C22×C3 | C3×SL(2,ℤ/4) |
| cosocle | #127⋊C9 | C2×C3 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 | |
148 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2,3^2,2^4|a^2=c,d^3=e,f^2=h,g^2=i,d^a=d^2e^2,e^a=e^2,f^a=fghi,g^a=gh,h^a=hi,d^b=de,f^c=fh,g^c=gi,f^d=gh,g^d=fg,h^d=i,i^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2826);