Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Sylow | C26 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C22⋊(C22⋊A4) |
| commutator, socle, cosocle | C26×C3 | C32 |
| Fitting, prime radical | C26×C32 | C3 |
| hypercenter | C32 | C22⋊(C22⋊A4) |
| hypocenter | C26 | 3−1+2 |
| LCS | G ▷ C26×C3 ▷ C26 | |
| UCS | C1 ◁ C3 ◁ C32 | |
5914 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
3^3,2^6|a^3=c,b^a=bc,d^a=de,e^a=d,f^a=fg,g^a=f,h^a=hi,i^a=h200 conjugacy classes — view the full character table.
| maximal subgroup of | #72252810, #72252823, A4×#88907 |
| maximal normal subgroup of | #72252810, #72252823, A4×#88907 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46179);