Finite Group DatabasebetaFinite Group Databasebeta| order | 22599 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C31 · Schur multiplier (H2) = C33 |
| direct product | C31×#10624723 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C31 | C3×H(3,𝔽3) |
| commutator | C32 | C32×C9×C31 |
| Frattini, cosocle | C33 | C33×C31 |
| prime radical | C33×C31 | C33 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C31 ◁ C32×C9×C31 ◁ G | |
250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 7533 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,5,4,3,6,9,8,7)(10,11,15,13,17,25,14,18,24)(12,19,23,21,28,34,22,16,26)(20,30,33,32,27,36,31,29,35)(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);(10,12,20,13,21,32,14,22,31)(11,16,27,17,19,29,18,28,30)(15,23,33,25,34,36,24,26,35);(1,3,7,4,8,5,9,2,6)(12,21,22)(16,19,28)(20,31,32)(23,34,26)(27,30,29)(33,35,36)3255 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22599, 283);