Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C43 · Schur multiplier (H2) = C3 |
| direct product | C43×#10624490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C43 | #732 |
| commutator, Frattini, cosocle | C32×C9 | C32×C43 |
| prime radical | C33×C43 | H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C43 ◁ C33×C43 ◁ C32×C9×C43 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 79 points:
(2,4,6)(10,11,14,13,18,24,22,17,30)(12,19,23,21,16,29,32,26,34)(15,25,31,28,33,36,27,35,20)(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);(1,2,5)(3,6,9)(4,8,7)(11,15,26,18,28,19,17,27,16)(14,23,25,30,34,35,24,29,33)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 50);