Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Fitting | D4×C32⋊C9 ⋊ C2 |
| Sylow | C32⋊C9 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#4169 |
| commutator, cosocle | C2×C3×C9 | C23×C3 |
| Frattini, socle | C2×C32 | C22×C3×D3 |
| Fitting | D4×C32⋊C9 | C2 |
| hypercenter | D4 | #4169 |
| hypocenter | C3×C9 | C3×#41 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ D4 | |
332 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 82 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:
2^3,3,2,3^3|b^2=e,f^3=h,c^a=ce,f^a=f^2h^2,g^a=g^2,h^a=h^2,c^b=ce,f^d=fg105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1058);