Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C43 · Schur multiplier (H2) = C9 |
| direct product | C43×#10624537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | #6879 |
| commutator, Frattini, cosocle | C92 | C32×C43 |
| prime radical | C43×#6879 | C3 |
| LCS | G ▷ C92 ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C32×C43 ◁ C3×C9×C43 ◁ C92×C43 ◁ G | |
60 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)(3,11,18)(4,9,19)(5,10,22)(7,23,13)(8,27,42)(12,35,44)(14,34,47)(15,30,48)(16,28,20)(17,31,21)(24,55,65)(25,52,39)(26,45,40)(29,50,36)(32,51,37)(33,53,68)(38,63,72)(41,58,46)(43,60,49)(54,69,59)(56,71,67)(57,74,80)(61,70,66)(62,73,64)(75,81,77)(76,79,78)(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);(1,3,12,36,64,79,4,13,37,5,14,38,15,39,65,16,40,66,17,41,67,42,68,80,43,69,81)(2,7,24,8,11,32,28,33,35,10,26,57,29,34,61,60,62,63,31,59,78,30,58,77,9,25,56)(6,18,44,50,73,76,19,23,51,22,47,72,48,52,55,20,45,70,21,46,71,27,53,74,49,54,75)3827 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 97);