Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C32×C9 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32⋊D9 |
| commutator, cosocle | C9×C3⋊Dic3 | C22 |
| Frattini | C2×C3 | D32⋊D3 |
| Fitting | C2×C32×C9 | D4 |
| socle | C2×C33 | #67 |
| hypocenter | C32×C9 | D8 |
| LCS | G ▷ C9×C3⋊Dic3 ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
154 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^4|c^2=d,g^3=h,b^a=bc,c^a=cd,e^a=e^2,f^a=e^2f,g^a=g^2h^2,h^a=h^2,c^b=cd,e^b=e^2f^2,e^c=e^2,f^c=f^275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1966);