Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32×C9 · Schur multiplier (H2) = C23×C33 |
| direct product | C2×C32×D4×C9 |
| unambiguous semidirect product | C4⋊(C22×C32×C9) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C9 | C22 |
| commutator | C2 | C23×C32×C9 |
| Frattini, cosocle | C2×C3 | C23×C33 |
| socle | C22×C33 | C22×C3 |
| prime radical | C2×C33×D4 | C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C9 ◁ G | |
1350 conjugacy classes of subgroups; only the 950 normal subgroups are shown.
This group has 1350 conjugacy classes of subgroups — too many to draw. Only its 950 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(7,8,9);(3,4,5,6)(10,11,12);(4,6)(13,14,15,16,17,18,19,20,21)810 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2693);