Finite Group DatabasebetaFinite Group Databasebeta| order | 31347 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C43 · Schur multiplier (H2) = C9 |
| direct product | C43×#10624528 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C43 | #732 |
| commutator | C3×C9 | C3×C9×C43 |
| Frattini, cosocle | C32×C9 | C32×C43 |
| prime radical | C33×C43 | H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C43 ◁ C33×C43 ◁ C32×C9×C43 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 3483 |
A minimal faithful permutation representation — 2 generators on 79 points:
(1,2,4,3,5,7,6,8,9)(10,11,15)(12,18,14)(13,17,30)(16,28,35)(19,34,36)(20,26,24)(21,27,25)(22,33,32)(23,31,29)(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);(10,12,19,14,21,29,25,11,16)(13,22,15,24,17,18,32,20,27)(23,31,33,28,35,30,34,36,26)2451 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(31347, 88);