Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C23×C4×C9 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C9⋊S4 |
| commutator, cosocle | C2×C9×A4 | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C23×C4×C9 | D3 |
| socle | C23×C3 | C22×C3⋊D3 |
| hypercenter | C2×C4 | C9⋊S4 |
| hypocenter | C9×A4 | #41 |
| LCS | G ▷ C2×C9×A4 ▷ C9×A4 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
673 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 673 conjugacy classes of subgroups — too many to draw. Only its 89 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|b^2=d,f^3=g,c^a=cd,e^a=e^2,f^a=f^2g^2,g^a=g^2,h^a=hi,h^e=i,i^e=hi156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30515);