Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#19837 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | D42 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| prime radical | #459×C89 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C42×C89 ◁ G | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 113 points:
(2,5)(3,8)(6,7)(9,10,14,17)(11,15,13,21)(12,19,20,23)(16,22,24,18)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,2,4,7)(3,6,8,5)(9,11,14,13)(10,15,17,21)(12,18,20,22)(16,23,24,19);(9,12,14,20)(10,16,17,24)(11,18,13,22)(15,23,21,19);(2,6)(3,8)(5,7)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 12825);