Finite Group DatabasebetaFinite Group Databasebeta| order | 16767 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C23 · Schur multiplier (H2) = C33 |
| direct product | C23×#10624505 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C23 | #6866 |
| commutator | C33 | C3×C9×C23 |
| Frattini, cosocle | C34 | C32×C23 |
| prime radical | C32×C23×H(3,𝔽3) | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C23 ◁ C33×C23 ◁ C34×C23 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 621 |
A minimal faithful permutation representation — 2 generators on 50 points:
(1,2,5)(3,9,18,11,16,25,23,6,13)(4,7,14)(8,17,12)(10,20,27,21,19,26,22,15,24)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50);(1,3,10)(2,6,15)(4,11,22)(5,13,24)(7,9,19)(8,16,20)(12,23,21)(14,18,26)(17,25,27)1311 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(16767, 65);