Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C34×C43 · Schur multiplier (H2) = C36 |
| direct product | C43×#10624930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C43 | C32×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34×C43 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C43 ◁ C3×H(3,𝔽3)×C43 ◁ G | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 70 |
| transitive degree | μt(G) = 1161 |
A minimal faithful permutation representation — 4 generators on 70 points:
(1,2,5)(3,7,13)(4,6,14)(8,18,12)(9,16,21)(10,17,22)(11,15,23)(19,24,26)(20,25,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(2,6,8)(7,15,17)(16,24,25);(1,3,9)(2,7,16)(4,11,19)(5,13,21)(6,15,24)(8,17,25)(10,20,12)(14,23,26)(18,22,27);(3,10,11)(7,17,15)(9,19,20)(13,22,23)(16,24,25)(21,26,27)4515 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 490);