Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C43 · Schur multiplier (H2) = C34 |
| direct product | C3×C43×#6867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C43 | H(3,𝔽3) |
| commutator | C32 | C32×C9×C43 |
| Frattini, cosocle | C33 | C33×C43 |
| prime radical | C34×C43 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33×C43 ◁ C34×C43 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 82 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 3 generators on 82 points:
(1,2,3)(40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82);(4,5,7,6,8,10,9,11,12)(13,14,17)(15,21,25)(16,20,26)(18,27,23)(19,30,24)(22,32,35)(28,38,39)(29,37,34)(31,36,33);(13,15,22,16,23,33,24,34,39)(14,18,28,20,29,32,19,21,31)(17,25,35,26,27,36,30,37,38)6579 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 246);