Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33×C4 · Schur multiplier (H2) = C23×C32 |
| direct product | C22×C3×C4×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32×C4 | C32 |
| commutator | C3 | C22×C33×C4 |
| Frattini, cosocle | C2×C3 | C23×C33 |
| socle | C23×C32 | C2×C32 |
| prime radical | C23×C33 | C2×C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C22×C32×C4 ◁ G | |
1026 conjugacy classes of subgroups; only the 864 normal subgroups are shown.
This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 864 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(9,10,11);(3,4)(12,13,15,14,17,19,18,16,20);(5,6,7,8)(13,16,17)(15,19,20)528 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2691);