Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C92×C31 · Schur multiplier (H2) = C32 |
| direct product | C31×#10624473 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C31 | C9⋊C9 |
| commutator | C9 | C92×C31 |
| Frattini, cosocle | C32×C9 | C32×C31 |
| socle | C3×C31 | #6855 |
| prime radical | C33×C31 | 3−1+2 |
| LCS | G ▷ C9 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C31 ◁ C32×C9×C31 ◁ G | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 112 |
| transitive degree | μt(G) = 2511 |
A minimal faithful permutation representation — 2 generators on 112 points:
(1,2,7)(3,12,38,15,41,58,17,29,65)(4,10,34)(5,9,31)(6,11,25)(8,26,51,28,59,47,30,60,48)(13,43,72,45,70,80,50,77,57)(14,42,53,16,40,54,49,66,23)(18,35,67)(19,33,55)(20,36,24)(21,32,68)(22,37,56)(27,61,74,62,69,46,64,81,44)(39,52,73,63,78,76,71,79,75)(82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112);(1,3,13,5,16,46,21,51,76,22,15,50,20,49,74,18,47,73,6,17,45,4,14,44,19,48,75)(2,8,27,10,29,63,35,66,77,37,28,64,9,12,39,33,42,70,11,30,62,36,41,71,32,40,43)(7,23,52,24,26,57,55,58,61,56,53,79,34,59,80,68,65,81,25,54,78,31,60,72,67,38,69)3255 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 33);