Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C35×C43 · Schur multiplier (H2) = C311 |
| direct product | C33×H(3,𝔽3)×C43 |
| unambiguous semidirect product | C32⋊(C34×C43) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34×C43 | C32 |
| commutator, Frattini, cosocle | C3 | C35×C43 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34×C43 ◁ G | |
11744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 10449 |
A minimal faithful permutation representation — 5 generators on 61 points:
(1,2,3)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61);(4,5,6);(7,8,9);(10,11,13)(12,15,17)(14,18,16);(11,14,15)(13,17,18)12771 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 498);