Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C33×C16 ⋊ C22 |
| Sylow | C33 ⋊ C2×M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×#1113 |
| commutator, socle | C2×C33 | C22×C8 |
| Fitting | C33×C16 | C22 |
| prime radical | C2×#1113 | C8 |
| cosocle | C33×C8 | C23 |
| hypercenter | C16 | #1113 |
| hypocenter | C33 | C2×M32 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C8 ◁ C16 | |
450 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|c^2=d,d^2=e,e^2=f,g^a=g^2,h^a=h^2,c^b=cf,g^b=g^2,i^b=i^2192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 27527);