Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C27×C31 · Schur multiplier (H2) = C3 |
| direct product | C31×3−1+2⋊C27 |
| unambiguous semidirect product | 3−1+2⋊(C27×C31) |
| 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 | |
126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 7533 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,5)(3,7,4)(6,9,8)(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36)(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);(1,3,7,4,6,9,8,2,5)4743 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 8);