Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C26 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | A4×C22⋊A4 |
| commutator, Fitting, socle, cosocle | C26×C3 | C32 |
| prime radical | C24×C3×A4 | C3 |
| hypocenter | C26 | 3−1+2 |
| LCS | G ▷ C26×C3 ▷ C26 | |
| UCS | C1 ◁ C3 | |
1013 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 1013 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
3^3,2^6|b^3=c,b^a=bc,h^a=hi,i^a=h,d^b=de,e^b=d,f^b=fg,g^b=f80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252811, #72252814, A4×#88907, … |
| maximal normal subgroup of | #72252811, #72252814, A4×#88907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46130);