Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60315 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C25 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C2×C4 | C25×C89 |
| socle | C22×C89 | C24×C4 |
| prime radical | C89×#3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ G | |
1816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 129 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,16,25,29)(11,15,21,28,19,26,33,37)(12,22,23,36,24,17,30,31)(20,34,35,40,32,27,38,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,28)(22,34)(23,35)(24,32)(25,33)(29,37)(30,38)(31,39)(36,40);(3,8)(5,7)(10,16)(11,19)(17,22)(18,29)(20,32)(21,33)(31,36)(35,38);(2,6)(5,7)(11,19)(15,26)(20,32)(21,33)(27,34)(28,37)(35,38)(39,40);(9,12,13,23,14,24,25,30)(10,17,18,31,16,22,29,36)(11,20,21,35,19,32,33,38)(15,27,28,39,26,34,37,40)6764 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53303);