Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C43 · Schur multiplier (H2) = C39 |
| direct product | C3×C43×C32⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C43 | C33 |
| commutator, Frattini, cosocle | C32 | C34×C43 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33×C43 ◁ G | |
3824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,3)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(4,5,7)(6,9,11)(8,12,10)(13,14,16)(15,18,20)(17,21,19);(5,8,9)(7,11,12);(14,17,18)(16,20,21)6579 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 425);