Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C26 |
| direct product | C89×#33523 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C2×#576×C89 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×#41×C89 ◁ C2×#131×C89 ◁ G | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2)(3,4)(5,6,10,15)(7,12,16,26)(8,11,21,25)(9,22,20,14)(13,29,24,19)(17,23,30,34)(18,33,31,28)(27,36,35,32)(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);(6,11)(8,19)(9,20)(12,23)(13,14)(15,25)(17,32)(18,31)(21,29)(22,24)(26,34)(27,28)(30,36)(33,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,16)(11,23)(13,27)(14,28)(15,26)(19,32)(20,31)(21,30)(22,33)(24,35)(25,34)(29,36);(7,16)(12,26)(17,30)(18,31)(23,34)(27,35)(28,33)(32,36)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26511);