Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C27 ⋊ #398 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D27 |
| commutator, cosocle | C23×C27 | C23 |
| Frattini | C23×C9 | C22×D3 |
| Fitting | C27×#95 | C2 |
| socle, prime radical | C23×C3 | C22×D9 |
| hypercenter | #95 | D27 |
| hypocenter | C27 | #398 |
| LCS | G ▷ C23×C27 ▷ C27 | |
| UCS | C1 ◁ C23 ◁ #95 | |
300 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1728 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=e,c^2=ef,g^3=h,h^3=i,b^a=be,c^a=cef,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2,c^b=cd282 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 230);