Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C33 ⋊ #538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D33 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C23×C33 | C23 |
| hypercenter | C23 | D33 |
| hypocenter | C33 | #538 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
1282 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|c^2=e,c^a=cf,d^a=de,h^a=h^2,c^b=cf,d^b=def,g^b=g^2,d^c=de,i^c=i^2102 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 37865);