Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624502 |
| subgroup | G/H | |
|---|---|---|
| center | C27×C43 | H(3,𝔽3) |
| commutator | C32 | C3×C27×C43 |
| Frattini, cosocle | C3×C27 | C32×C43 |
| socle | C3×C43 | C32⋊C27 |
| prime radical | C33×C43 | C27 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C27×C43 ◁ C3×C27×C43 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 124 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 124 points:
(1,2,6,3,9,18,11,25,37,4,10,19,12,26,38,28,48,59,14,27,39,30,49,60,50,68,74,5,7,17,13,20,35,29,40,56,15,21,36,31,41,57,51,61,72,33,42,58,53,62,73,69,75,80,16,8,22,32,23,43,52,45,63,34,24,44,54,46,64,70,66,76,55,47,65,71,67,77,79,78,81)(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);(2,7,8)(9,20,23)(10,21,24)(25,40,45)(26,41,46)(27,42,47)(48,61,66)(49,62,67)(68,75,78)6579 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 62);