Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C23 |
| direct product | C89×#9308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #386 |
| commutator | C42 | C22×C4×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| prime radical | C23×C89 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 129 points:
(2,6)(3,4)(7,8)(9,10)(11,15,20,31)(12,22)(13,26)(14,17)(16,24)(18,28)(19,35,23,40)(21,38,37,30)(25,34)(27,33)(29,39,32,36)(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);(1,2,4,6,5,7,3,8)(9,11)(10,15,17,31)(12,23)(13,21)(14,20)(16,32,33,29)(18,30,26,38)(19,25)(22,40,34,35)(24,39)(27,36)(28,37);(1,3,5,4)(2,6,7,8)(9,12,14,25)(10,16,17,33)(11,19,20,23)(13,27,28,24)(15,29,31,32)(18,34,26,22)(21,39,37,36)(30,40,38,35)3293 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 2296);