Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C31 · Schur multiplier (H2) = C34 |
| direct product | C31×#10624722 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C31 | C3×H(3,𝔽3) |
| commutator | C32 | C32×C9×C31 |
| Frattini, cosocle | C33 | C33×C31 |
| prime radical | C3×C31×#732 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C31 ◁ C32×C9×C31 ◁ G | |
304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 7533 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,5,4,3,6,9,8,7)(10,11,14)(12,18,23)(13,17,24)(15,25,21)(16,28,22)(19,30,33)(20,27,34)(26,36,32)(29,35,31)(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);(10,12,19,22,20,32,13,21,31)(11,15,26,16,18,29,17,27,30)(14,23,33,28,34,36,24,25,35);(1,3,7,4,8,5,9,2,6)(12,20,21)(15,18,27)(19,31,32)(23,34,25)(26,30,29)(33,35,36)3255 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 282);