Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C33 ⋊ #335 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #6295 |
| commutator | C33×C4 | C2×C8 |
| Frattini, hypercenter | C42 | #1113 |
| Fitting, cosocle | C33×C42 | C22 |
| socle, prime radical | C22×C33 | C4⋊C4 |
| hypocenter | C33 | #335 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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|a^2=d,b^2=c,c^2=e,d^2=f,b^a=bd,g^a=g^2,i^a=i^2,d^b=df,g^b=g^2,h^b=h^2180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11771);