Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#62174 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C89×#3524 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ #529×C89 ◁ G | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 137 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,28)(20,32)(21,26)(22,27)(23,36)(24,31)(25,33)(29,44)(30,40)(34,41)(35,39)(37,42)(38,45)(43,47)(46,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,29)(21,35)(22,31)(23,30)(25,38)(26,40)(27,28)(32,48)(33,47)(34,46)(36,39)(37,45)(41,44)(42,43);(17,20,22,34)(18,25,27,42)(19,29,31,46)(21,33,23,37)(24,38,28,43)(26,41,36,32)(30,45,35,47)(39,48,40,44);(3,10)(6,9)(11,16)(14,15)(17,21,22,23)(18,26,27,36)(19,30,31,35)(20,33,34,37)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,27)(19,31)(25,42)(26,36)(29,46)(30,35)(32,41)(45,47)4361 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 55162);