Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×D4×H(3,𝔽3) ⋊ C22 |
| Sylow | H(3,𝔽3) ⋊ C2×H(5,𝔽2) |
3309 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^6,3^3|g^a=g^2,i^a=i^2,e^b=ef,h^b=h^2,i^b=i^2,d^c=df,h^g=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72220460, H(5,𝔽2)×((C3⋊D3)⋊C3)⋊C2, #72252959, … |
| maximal normal subgroup of | H(5,𝔽2)×((C3⋊D3)⋊C3)⋊C2, #72252959, #72252984 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30806);