Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C22×C3×Dic27 |
| derived | C27 ⋊ C22×C3×C4 |
| Sylow | C3×C27 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D27 |
| commutator, hypocenter | C27 | C22×C3×C4 |
| Frattini | C2×C9 | C22×C3×D3 |
| Fitting | C23×C3×C27 | C2 |
| socle, prime radical | C23×C32 | D9 |
| cosocle | C2×C27 | C23×C3 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C23×C3 | |
264 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
360 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1113);