Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C23×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C23×C32 |
| prime radical | C23×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C32 ◁ G | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2)(7,8,11)(9,15,19)(10,14,20)(12,21,17)(13,24,18)(16,26,29)(22,32,33)(23,31,28)(25,30,27);(3,4)(7,9,16,10,17,27,18,28,33)(8,12,22,14,23,26,13,15,25)(11,19,29,20,21,30,24,31,32);(5,6)136 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 347);