Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C27×C4 |
| direct product | C89×#60624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C89×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G | |
2038 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,16)(13,18)(14,29)(15,20)(17,24)(19,22)(23,35)(25,30)(26,31)(27,33)(28,39)(32,37)(34,36)(38,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(2,5)(6,8)(9,11,22,26,15,24,14,12)(10,16,29,17,20,31,19,21)(13,23,36,38,27,37,28,25)(18,30,39,32,33,40,34,35);(1,3,4,7)(2,6,5,8)(9,12,14,24,15,26,22,11)(10,17,19,16,20,21,29,31)(13,25,28,37,27,38,36,23)(18,32,34,30,33,35,39,40);(9,13)(10,18)(11,23)(12,25)(14,28)(15,27)(16,30)(17,32)(19,34)(20,33)(21,35)(22,36)(24,37)(26,38)(29,39)(31,40);(13,27)(18,33)(23,37)(25,38)(28,36)(30,40)(32,35)(34,39)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53612);