Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #10593838⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | A4×#157 |
| commutator, hypocenter | C32×Q8 | C3×C8 |
| Fitting | C32×#41 | C3×C4 |
| socle | C2×C32 | C8×A4 |
| prime radical | C32×#41⋊C3 | C4 |
| cosocle | #64157 | C2×C3 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C4 | |
122 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2,3,2^4,3^2,2|a^2=c,c^2=d,d^2=i,e^2=i,f^2=i,g^a=gh,h^a=gh^2,e^b=ef,f^b=e,g^c=g^2,h^c=h^2,f^e=fi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31908);