Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C29 |
| direct product | C89×#61064 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ #525×C89 ◁ G | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 125 points:
(1,2)(3,4)(5,6,11,16)(7,17,18,31)(8,13,23,30)(9,15,10,14)(12,24,26,20)(19,33,34,28)(21,36,35,27)(22,32,25,29)(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);(5,7,11,18)(6,12,16,26)(8,21,23,35)(9,20,10,24)(13,28,30,33)(14,17,15,31)(19,25,34,22)(27,32,36,29);(2,4)(7,18)(12,26)(17,31)(19,34)(20,24)(21,35)(27,36)(28,33);(5,8)(6,13)(7,19)(9,22)(10,25)(11,23)(12,27)(14,29)(15,32)(16,30)(17,33)(18,34)(20,35)(21,24)(26,36)(28,31);(5,9,11,10)(6,14,16,15)(7,20,18,24)(8,22,23,25)(12,17,26,31)(13,29,30,32)(19,35,34,21)(27,33,36,28)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 54052);