Finite Group DatabasebetaFinite Group Databasebeta| order | 189 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 67392 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C33 |
| abelian | C33×C7 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 189 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,3)(10,11,12,13,14,15,16);(4,5,6);(7,8,9)189 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(189, 13);