Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C31 · Schur multiplier (H2) = C3 |
| direct product | C31×#10624453 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C31 | H(3,𝔽3) |
| commutator | C33 | C3×C9×C31 |
| Frattini, cosocle | C32×C9 | C32×C31 |
| socle | C32×C31 | C32⋊C9 |
| prime radical | C3×H(3,𝔽3)×C31 | C9 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C31 ◁ C32×C9×C31 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 7533 |
A minimal faithful permutation representation — 2 generators on 67 points:
(2,4,6)(10,11,14,12,17,22,18,26,30,13,15,21,19,23,29,27,31,35,20,16,24,28,25,32,34,33,36)(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);(1,2,5)(3,6,9)(4,8,7)(11,15,16)(17,23,25)(26,31,33)3255 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 13);