Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D3×C3⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived, Sylow | C32×C9 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×C3⋊D9 |
| commutator, hypocenter | C32×C9 | C2×C8 |
| Frattini | C3×C4 | D3×C3⋊D3 |
| Fitting, cosocle | C32×C4×C9 | C22 |
| socle | C2×C33 | C4×D3 |
| prime radical | C2×C3⋊(C3⋊D9) | C4 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C4 | |
340 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 78 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:
2^4,3^4|a^2=c,b^2=c,c^2=d,f^3=h,g^a=g^2,e^b=e^2,f^b=f^2h^2,h^b=h^2180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1352);