Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C3×C9 |
| direct product | C43×#10624513 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6867 |
| commutator | C3×C9 | C3×C9×C43 |
| Frattini, cosocle | C32×C9 | C32×C43 |
| prime radical | C33×C43 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C33×C43 ◁ C32×C9×C43 ◁ G | |
96 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)(3,11,31,14,34,53,15,35,63)(4,16,21)(5,10,22)(7,23,40,25,54,72,26,55,73)(8,27,18)(9,30,19)(12,36,52,38,68,77,43,69,78)(13,39,49,41,67,50,42,33,20)(17,45,51)(24,56,71,57,62,74,58,79,37)(28,61,75)(29,60,47)(32,64,80,65,48,70,66,76,81)(44,59,46)(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);(1,3,12,19,14,43,5,15,38)(2,7,24,10,26,57,9,25,58)(4,13,37,47,41,74,18,42,71)(6,20,48)(8,11,32,29,35,65,16,34,66)(17,40,70,75,72,80,46,73,81)(21,23,52)(22,50,76)(27,55,77)(28,33,36,45,67,68,44,39,69)(30,49,64)(31,62,59)(51,53,56)(54,78,60)(61,63,79)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 73);