Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34×C4 · Schur multiplier (H2) = C36 |
| direct product | C34×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C34×C4 | C22 |
| commutator | C2 | C2×C34×C4 |
| Frattini, cosocle | C4 | C22×C34 |
| socle | C2×C34 | C2×C4 |
| prime radical | C22×C34 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C34×C4 ◁ G | |
2120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 648 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3)(13,14,15,16,17,18,19,20);(4,5,6)(14,18)(16,20);(7,8,9);(10,11,12)810 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2885);