Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C92×C31 · Schur multiplier (H2) = C92 |
| direct product | C31×H(3,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center | C9×C31 | C92 |
| commutator | C9 | C92×C31 |
| Frattini, cosocle | C32×C9 | C32×C31 |
| socle | C3×C31 | #6855 |
| prime radical | C33×C31 | H(3,𝔽3) |
| LCS | G ▷ C9 ▷ C1 | |
| UCS | C1 ◁ C9×C31 ◁ G | |
174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 112 |
| transitive degree | μt(G) = 2511 |
A minimal faithful permutation representation — 2 generators on 112 points:
(1,2,6)(3,10,29,13,32,57,14,33,58)(4,8,19)(5,9,21)(7,22,37,24,50,68,25,51,69)(11,34,64,36,65,49,40,71,76)(12,31,44,38,61,45,39,62,18)(15,26,46)(16,27,48)(17,28,20)(23,52,70,53,77,35,54,78,67)(30,59,80,60,43,66,63,73,81)(41,55,75)(42,56,47)(72,79,74)(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);(1,3,11)(2,7,23,9,25,53,28,24,54)(4,12,35)(5,14,36)(6,18,43,20,44,59,21,45,73)(8,10,30,27,33,60,56,32,63)(13,40,17)(15,37,66)(16,39,67)(19,22,49,47,50,64,48,51,76)(26,31,34,55,62,65,79,61,71)(29,52,74,57,78,75,58,77,46)(38,70,42)(41,69,81)(68,80,72)3255 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 24);