Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #336 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C23 | D3×#157 |
| commutator | C2×C33 | C4×C8 |
| Fitting, socle, prime radical | C23×C33 | C2×C4 |
| cosocle | C22×C3×C3⋊Dic3 | C22 |
| hypocenter | C33 | #336 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
560 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=c,b^2=d,c^2=f,b^a=be,h^a=hi,i^a=hi^2,g^b=g^2,h^c=h^2,i^c=i^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 20910);