Finite Group DatabasebetaFinite Group Databasebeta| order | 81 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 24261120 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C36 |
| abelian | C34 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,3);(4,5,6);(7,8,9);(10,11,12)81 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(81, 15);