Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C89 · Schur multiplier (H2) = C22×C42 |
| direct product | C89×#10711 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C89 | C22⋊C4 |
| commutator | C22 | C43×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C22×C89 | C4×C22⋊C4 |
| prime radical | C24×C89 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C89 ◁ C23×C8×C89 ◁ G | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 125 points:
(5,6)(7,15,17,11)(8,12)(9,14)(10,13)(16,27,29,22)(18,28,31,23)(19,25)(20,24)(21,26)(30,35,36,33)(32,34)(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);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,22)(13,15)(14,23)(19,30)(20,29)(21,31)(24,27)(25,33)(26,28)(32,36)(34,35);(5,8,9,19,10,20,21,32)(6,12,14,25,13,24,26,34)(7,16,18,30,17,29,31,36)(11,22,23,33,15,27,28,35)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 3699);