Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C33 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11722004 |
| commutator, Fitting, cosocle | C33×C16 | C22 |
| Frattini, hypercenter | C16 | D3×C3⋊D3 |
| socle | C2×C33 | D16 |
| hypocenter | C33 | D32 |
| LCS | G ▷ C33×C16 ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
252 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^3|c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,i^a=i^2,c^b=cd,d^b=de,e^b=ef,g^b=g^2,h^b=h^2138 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11553);