Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#24804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | #525×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 125 points:
(5,6,12,16,11,18,10,17)(7,15,9,27,21,29,20,13)(8,22,24,14,26,32,25,31)(19,33,34,30,35,28,23,36)(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);(1,2)(3,4)(5,7,10,20,11,21,12,9)(6,13,16,15,18,27,17,29)(8,19,24,34,26,35,25,23)(14,28,22,30,31,33,32,36);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,26)(12,25)(13,28)(15,30)(16,22)(17,32)(18,31)(20,34)(21,35)(27,33)(29,36);(5,9,12,21,11,20,10,7)(6,15,17,13,18,29,16,27)(8,23,25,35,26,34,24,19)(14,30,32,28,31,36,22,33)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 17792);