Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C43 · Schur multiplier (H2) = C33 |
| direct product | C43×#10624837 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C43 | #732 |
| commutator, Frattini, cosocle | C3×C9 | C33×C43 |
| socle | C3×C43 | C3×#732 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C43 ◁ C3×C9×C43 ◁ C32×C9×C43 ◁ G | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 1161 |
A minimal faithful permutation representation — 3 generators on 70 points:
(1,2,6)(3,8,18)(4,14,17)(5,11,19)(7,20,15)(9,23,12)(10,24,16)(13,22,26)(21,27,25)(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);(2,7,21,11,8,14,10,22,9)(6,17,20,24,27,26,19,23,18);(1,3,12,5,13,25,16,15,4)(2,8,9,11,22,21,10,7,14)(6,18,23,19,26,27,24,20,17)4515 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 397);