Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) order 209952 |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C35 |
| direct product | C3×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | C32 |
| commutator, Frattini, cosocle | C3 | C34 |
| socle | C32 | C33 |
| prime radical | C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3);(4,5,8)(6,10,16)(7,11,15)(9,17,14)(12,19,24)(13,20,23)(18,25,22)(21,27,29)(26,30,28);(5,9,11)(8,15,17)(10,18,20)(16,23,25)(19,26,27)(24,29,30);(4,6,12,7,13,21,14,22,28)(5,10,19,11,20,27,9,18,26)(8,16,24,15,23,29,17,25,30)99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 64);