Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C32 |
| direct product | C43×#10624512 |
| 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)(2,7,24,9,25,58,10,26,57)(4,13,37)(5,15,38)(6,20,48,22,50,76,30,49,64)(8,11,32,16,34,66,29,35,65)(14,43,19)(17,40,70)(18,42,71)(21,23,52,27,55,77,60,54,78)(28,33,36,44,39,69,45,67,68)(31,62,61,63,79,51,53,56,59)(41,74,47)(46,73,81)(72,80,75)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 72);