Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9×C43 · Schur multiplier (H2) = C35 |
| direct product | C9×C43×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C92×C43 | C32 |
| commutator | C3 | C33×C9×C43 |
| Frattini, cosocle | C32 | C34×C43 |
| socle | C32×C43 | C34 |
| prime radical | C3×H(3,𝔽3)×C43 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C92×C43 ◁ G | |
986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 4 generators on 79 points:
(1,2,4,3,5,7,6,8,9)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(10,11,14)(12,16,22)(13,17,21)(15,23,20)(18,25,30)(19,26,29)(24,31,28)(27,33,35)(32,36,34);(11,15,17)(14,21,23)(16,24,26)(22,29,31)(25,32,33)(30,35,36);(10,12,18,13,19,27,20,28,34)(11,16,25,17,26,33,15,24,32)(14,22,30,21,29,35,23,31,36)12771 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 421);