Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60713 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C89×#3058 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G | |
1878 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)(11,20)(12,23)(13,16)(14,18)(15,29)(17,26)(19,28)(21,35)(22,31)(24,37)(25,38)(27,34)(30,39)(32,40)(33,36)(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);(2,5)(6,8)(9,11,14,13)(10,16,18,20)(12,21,25,24)(15,22,28,27)(17,30,32,33)(19,31,29,34)(23,37,38,35)(26,36,40,39);(9,12,15,26,14,25,28,40)(10,17,19,23,18,32,29,38)(11,21,22,36,13,24,27,39)(16,30,31,37,20,33,34,35);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,16,18,20)(12,24,25,21)(15,27,28,22)(17,30,32,33)(19,31,29,34)(23,37,38,35)(26,39,40,36);(11,13)(16,20)(21,24)(22,27)(30,33)(31,34)(35,37)(36,39)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53701);