Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×#72221927 (order 23328) · Inn(G) = C1 · Out(G) = C2×#72221927 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C33 |
| abelian | C32×C9 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C33 |
| socle, prime radical | C33 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3,4,5,6,7,8,9);(10,11,12);(13,14,15)81 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 11);