Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624507 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6866 |
| commutator | C33 | C3×C9×C43 |
| Frattini, cosocle | C32×C9 | C32×C43 |
| prime radical | C34×C43 | C9 |
| LCS | G ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C3×C9×C43 ◁ C32×C9×C43 ◁ G | |
232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 1161 |
A minimal faithful permutation representation — 2 generators on 70 points:
(1,2,5,3,8,14,10,18,23,4,7,15,11,17,22,20,25,27,12,6,13,9,16,21,19,24,26)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(2,6,7)(3,9,11)(5,13,15)(8,17,16)(10,20,19)(14,22,21)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 67);