Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#11156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #386 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| prime radical | C24×C89 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 125 points:
(5,6,10,16,11,14,9,15)(7,17,20,27,21,12,19,28)(8,22,23,30,24,31,25,13)(18,32,33,26,34,36,35,29)(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)(5,7,11,21)(6,12,14,17)(8,18,24,34)(9,20,10,19)(13,29,30,26)(15,28,16,27)(22,36,31,32)(23,35,25,33);(2,4)(5,8)(6,13,14,30)(7,18)(9,23)(10,25)(11,24)(12,26,17,29)(15,22,16,31)(19,33)(20,35)(21,34)(27,36,28,32)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4144);