Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C43 · Schur multiplier (H2) = C36 |
| direct product | C32×C43×#731 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C43 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34×C43 |
| prime radical | C32×H(3,𝔽3)×C43 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33×C43 ◁ C34×C43 ◁ G | |
1160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 76 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 4 generators on 76 points:
(1,2,3)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76);(4,5,6);(7,8,11)(9,15,19)(10,14,20)(12,21,17)(13,24,18)(16,26,29)(22,32,33)(23,31,28)(25,30,27);(7,9,16,10,17,27,18,28,33)(8,12,22,14,23,26,13,15,25)(11,19,29,20,21,30,24,31,32)6579 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 480);