Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9×C43 · Schur multiplier (H2) = C37 |
| direct product | C3×C9×H(3,𝔽3)×C43 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9×C43 | C32 |
| commutator | C3 | C33×C9×C43 |
| Frattini, cosocle | C32 | C34×C43 |
| socle | C33×C43 | C33 |
| prime radical | C32×H(3,𝔽3)×C43 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9×C43 ◁ G | |
1688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,3)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(4,5,7,6,8,10,9,11,12);(13,14,16)(15,18,20)(17,21,19);(14,17,18)(16,20,21)12771 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 419);