Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #10596689⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32×A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Fitting | C2×C32×Q8 | C22×C3 |
| socle | C2×C32 | C23×A4 |
| hypercenter | C22 | D32×A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2 ◁ C22 | |
895 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 895 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^2,3^2,2|b^2=i,e^2=i,f^2=i,c^a=ci,g^a=g^2,c^b=ci,h^b=h^2,e^d=ef,f^d=e,f^e=fi90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46954);