Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#11188 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #394 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| prime radical | C24×C89 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2)(3,4)(5,6,10,15,11,14,9,26)(7,16,19,32,20,12,18,28)(8,21,24,30,25,31,23,13)(17,29,22,27,35,36,34,33)(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);(5,7,11,20)(6,12,14,16)(8,22,25,34)(9,19,10,18)(13,29,30,36)(15,32,26,28)(17,23,35,24)(21,33,31,27);(2,4)(5,8)(6,13,14,30)(7,17)(9,23)(10,24)(11,25)(12,27,16,33)(15,21,26,31)(18,34)(19,22)(20,35)(28,36,32,29)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4176);