Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#33687 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C23×C89 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ #117×C89 ◁ #405×C89 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 161 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18)(11,21,23,16)(12,26,27,62)(14,32,30,61)(15,35,31,20)(17,40,41,28)(19,44,43,34)(22,50,51,60)(24,56,54,65)(25,39,55,49)(29,63,36,45)(33,46,64,42)(37,52,47,69)(38,58,48,70)(53,67,59,71)(57,66,72,68)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,37)(18,21)(19,38)(20,39)(26,60)(27,51)(28,52)(29,53)(30,54)(31,55)(32,65)(33,57)(34,58)(35,49)(36,59)(40,69)(41,47)(42,66)(43,48)(44,70)(45,67)(46,68)(50,62)(56,61)(63,71)(64,72);(9,12,13,27)(10,17,18,41)(11,22,23,51)(14,33,30,64)(15,36,31,29)(16,37,21,47)(19,45,43,63)(20,46,35,42)(24,57,54,72)(25,59,55,53)(26,61,62,32)(28,44,40,34)(38,67,48,71)(39,68,49,66)(50,65,60,56)(52,70,69,58);(1,3,4,7)(2,6,5,8)(11,23)(16,21)(22,51)(24,54)(25,55)(37,47)(38,48)(39,49)(50,60)(52,69)(53,59)(56,65)(57,72)(58,70)(66,68)(67,71)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26675);