Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C31 · Schur multiplier (H2) = C32 |
| direct product | C31×#10624446 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C31 | H(3,𝔽3) |
| commutator | C32 | C3×C27×C31 |
| Frattini, cosocle | C32×C9 | C32×C31 |
| socle | C32×C31 | C32⋊C9 |
| prime radical | C33×C31 | C3×C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C31 ◁ C32×C9×C31 ◁ G | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7533 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2,5)(3,9,13)(4,8,14)(6,15,11)(7,18,12)(10,20,23)(16,26,27)(17,25,22)(19,24,21)(28,29,32,30,33,38,35,40,45,31,34,39,36,41,46,43,48,51,37,42,47,44,49,52,50,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,10,12,22,27,4,11,21)(2,6,16,7,9,19,8,17,20)(5,13,23,18,25,26,14,15,24)4743 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 6);