Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 4478976 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C33 |
| abelian | C22×C32×C4×C9 |
| multiplicative | (ℤ/1729)ˣ, (ℤ/2223)ˣ, (ℤ/2331)ˣ, (ℤ/2457)ˣ, (ℤ/3458)ˣ, (ℤ/4446)ˣ, (ℤ/4662)ˣ, (ℤ/4914)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C3 | C23×C33 |
| socle, prime radical | C23×C33 | C2×C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 1296 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3,4)(9,10,11,12,13,14,15,16,17);(5,6)(18,19,20);(7,8)(21,22,23)1296 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2689);