Finite Group DatabasebetaFinite Group Databasebeta| order | 243 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | Aut(G) order 104976 |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C35 |
| extraspecial | 3−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C3 | C34 |
| prime radical | C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,5,4,8,15,13,7,19)(3,9,20,12,18,14,11,6,16)(10,22,26,23,17,27,24,21,25);(1,3,10)(2,6,17)(4,12,23)(5,14,25)(7,18,22)(8,9,21)(11,24,13)(15,16,26)(19,20,27);(3,11,12)(6,18,9)(10,23,24)(14,20,16)(17,21,22)(25,26,27);(2,7,8)(5,15,19)(6,18,9)(14,16,20)(17,22,21)(25,26,27)Polycyclic presentation:
3^5|a^3=e,b^a=be,d^a=de,c^b=ce83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(243, 66);