Finite Group DatabasebetaFinite Group Databasebeta| order | 693 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C23×C32×C5 (order 360) · Inn(G) = C1 · Out(G) = C23×C32×C5 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C693 |
| abelian | C7×C9×C11 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C3×C7×C11 |
| socle, prime radical | C3×C7×C11 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 693 |
A minimal faithful permutation representation — 1 generator on 27 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27)693 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(693, 2);