Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C43 · Schur multiplier (H2) = C36 |
| direct product | C3×C43×#6912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C43 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34×C43 |
| prime radical | C34×C43 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C43 ◁ C34×C43 ◁ G | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73);(4,5,8,7,11,18,16,10,22)(6,12,17,15,9,19,14,21,28)(13,24,27,25,23,29,26,20,30);(4,6,13,16,14,26,7,15,25)(5,9,20,10,12,23,11,21,24)(8,17,27,22,28,30,18,19,29);(6,14,15)(9,12,21)(13,25,26)(17,28,19)(20,24,23)(27,29,30)4515 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 487);