Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C31 · Schur multiplier (H2) = C32 |
| direct product | C31×#10624509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C31 | #6866 |
| commutator | C33 | C3×C9×C31 |
| Frattini, cosocle | C34 | C32×C31 |
| prime radical | C34×C31 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C31 ◁ C33×C31 ◁ C34×C31 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 58 |
| transitive degree | μt(G) = 837 |
A minimal faithful permutation representation — 2 generators on 58 points:
(1,2,5)(3,6,15,11,17,13,10,18,24)(4,7,14)(8,19,12)(9,20,25,22,26,27,21,16,23)(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);(1,3,9,4,11,21,12,10,22)(2,6,16,7,17,26,8,18,20)(5,13,23,14,24,27,19,15,25)1767 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 69);