Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C29 |
| direct product | C89×#60143 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C25 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C2×C4 | C25×C89 |
| socle | C22×C89 | C24×C4 |
| prime radical | C89×#3449 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ G | |
2706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 125 points:
(1,2)(3,4)(5,6,9,14,10,13,20,26)(7,11,17,22,16,23,29,33)(8,12,19,25,18,24,30,34)(15,21,28,32,27,31,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,21)(13,23)(14,22)(18,27)(19,28)(20,29)(24,31)(25,32)(26,33)(30,35)(34,36);(5,8,9,19,10,18,20,30)(6,12,14,25,13,24,26,34)(7,15,17,28,16,27,29,35)(11,21,22,32,23,31,33,36);(8,18)(12,24)(15,27)(19,30)(21,31)(25,34)(28,35)(32,36);(6,13)(7,16)(12,24)(14,26)(15,27)(17,29)(25,34)(28,35)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53131);