Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C33 |
| direct product | C43×#10624508 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6866 |
| commutator | C33 | C3×C9×C43 |
| Frattini, cosocle | C34 | C32×C43 |
| prime radical | C32×H(3,𝔽3)×C43 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C33×C43 ◁ C34×C43 ◁ G | |
350 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)(3,6,15,11,17,13,10,18,24)(4,7,14)(8,19,12)(9,20,25,22,26,27,21,16,23)(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);(1,3,9)(2,6,16)(4,11,21)(5,13,23)(7,17,26)(8,18,20)(10,22,12)(14,24,27)(15,25,19)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 68);