Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#33832 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C89×#3280 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×D4×C89 ◁ C8×D4×C89 ◁ G | |
686 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)(7,16,20,27)(8,21,24,30)(9,12,23,14)(11,26,29,18)(13,31,33,22)(17,34,35,28)(19,36,25,32)(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);(5,7,10,20)(6,11,15,29)(8,17,24,35)(9,25,23,19)(12,28,14,34)(13,32,33,36)(16,30,27,21)(18,31,26,22);(2,4);(6,12)(7,17)(8,22)(9,23)(11,27)(14,15)(16,29)(18,19)(20,35)(21,30)(24,31)(25,26)(28,32)(34,36)3827 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26820);