Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6866 |
| commutator | C33 | C3×C9×C43 |
| Frattini, cosocle | C32×C9 | C32×C43 |
| prime radical | C3×H(3,𝔽3)×C43 | C9 |
| LCS | G ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C3×C9×C43 ◁ C32×C9×C43 ◁ G | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 124 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 124 points:
(1,2,6,4,9,20,16,29,48,5,8,21,17,27,49,42,56,74,18,10,31,43,30,62,66,61,78)(3,7,22,13,24,47,37,54,72,14,26,19,38,55,45,41,60,70,12,25,46,15,28,57,40,59,73)(11,32,50,34,53,69,63,76,80,33,23,44,39,51,68,65,75,79,35,52,67,36,58,71,64,77,81)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124);(1,3,11)(2,7,23)(4,15,39)(5,14,33)(6,19,44)(8,26,52)(9,28,58)(10,25,32)(12,35,18)(13,36,17)(16,41,64)(20,47,71)(21,46,67)(22,50,31)(24,53,27)(29,60,76)(30,55,51)(34,43,38)(37,65,66)(40,63,42)(45,69,49)(48,73,80)(54,77,61)(56,59,75)(57,68,62)(70,81,78)(72,79,74)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 70);