Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D9×S4 |
| commutator, cosocle | C2×C9×A4 | C23 |
| Frattini | C2×C3 | C2×D3×S4 |
| Fitting | C22×C4×C9 | C2×D3 |
| socle | C23×C3 | C2×D32 |
| hypercenter | C4 | D9×S4 |
| hypocenter | C9×A4 | #41 |
| LCS | G ▷ C2×C9×A4 ▷ C9×A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
521 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^4,3^3,2^2|a^2=d,b^2=d,c^2=d,f^3=g,c^a=cd,f^a=f^2g^2,g^a=g^2,e^b=e^2,h^b=hi,h^e=i,i^e=hi105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30167);