Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C79×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
1084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2,8,14)(3,15,20,44)(4,21,24,41)(5,13,7,11)(6,26,29,39)(9,31,35,18)(10,30,37,22)(12,28,40,25)(16,45,49,61)(17,38,19,43)(23,54,56,36)(27,33,50,34)(32,55,60,64)(42,48,63,46)(47,51,58,59)(52,57,62,53)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(1,3)(2,9)(4,16)(5,17)(6,27)(7,19)(8,20)(10,32)(11,33)(12,38)(13,34)(14,35)(15,26)(18,25)(21,51)(22,46)(23,55)(24,49)(28,31)(29,50)(30,48)(36,61)(37,60)(39,44)(40,43)(41,59)(42,52)(45,54)(47,53)(56,64)(57,58)(62,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,32)(11,21)(12,36)(13,41)(14,37)(15,42)(17,46)(18,47)(19,48)(20,49)(25,53)(26,52)(27,55)(28,57)(29,56)(31,58)(33,51)(34,59)(35,60)(38,61)(39,62)(40,54)(43,45)(44,63)(50,64);(1,5,8,7)(2,11,14,13)(3,17,20,19)(4,22,24,30)(6,25,29,28)(9,33,35,34)(10,21,37,41)(12,26,40,39)(15,43,44,38)(16,46,49,48)(18,50,31,27)(23,53,56,57)(32,51,60,59)(36,52,54,62)(42,45,63,61)(47,64,58,55)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26530);