Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C31 · Schur multiplier (H2) = C33 |
| direct product | C31×#10624725 |
| subgroup | G/H | |
|---|---|---|
| center | C9×C31 | C3×H(3,𝔽3) |
| commutator | C32 | C32×C9×C31 |
| Frattini, cosocle | C3×C9 | C33×C31 |
| socle | C3×C31 | C9×H(3,𝔽3) |
| prime radical | C31×#6908 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9×C31 ◁ C3×C27×C31 ◁ G | |
216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 112 |
| transitive degree | μt(G) = 2511 |
A minimal faithful permutation representation — 3 generators on 112 points:
(1,2,6,4,10,19,14,27,40,5,7,18,15,21,39,34,43,62,16,9,25,35,26,49,56,50,69)(3,8,17,13,24,38,33,48,61,11,20,36,30,42,58,53,64,74,28,22,44,51,46,65,70,67,78)(12,23,37,32,47,60,55,68,76,29,41,57,52,63,73,71,77,80,31,45,59,54,66,75,72,79,81)(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);(2,7,9)(3,11,28)(8,22,20)(10,21,26)(12,31,29)(13,30,51)(17,36,44)(24,46,42)(27,43,50)(32,54,52)(33,53,70)(37,59,57)(38,58,65)(48,67,64)(55,72,71)(60,75,73)(61,74,78)(76,81,80);(1,3,12,4,13,32,14,33,55,5,11,29,15,30,52,34,53,71,16,28,31,35,51,54,56,70,72)(2,8,23,10,24,47,27,48,68,7,20,41,21,42,63,43,64,77,9,22,45,26,46,66,50,67,79)(6,17,37,19,38,60,40,61,76,18,36,57,39,58,73,62,74,80,25,44,59,49,65,75,69,78,81)3255 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 285);