Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33617 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×#466 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
642 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)(3,15)(4,22)(5,11)(6,26)(7,13)(8,14)(9,19)(10,23)(12,16)(17,45)(18,40)(20,41)(21,42)(24,54)(25,34)(27,38)(28,37)(29,36)(30,57)(31,43)(32,49)(33,50)(35,55)(39,46)(44,53)(47,52)(48,61)(51,62)(56,60)(58,63)(59,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(1,3,16,43,5,18,28,50,8,21,27,49,7,20,6,19)(2,9,26,41,13,32,38,42,14,33,37,40,11,31,12,15)(4,17,44,57,23,46,55,63,25,48,56,64,29,51,24,47)(10,30,53,45,22,52,54,62,36,59,60,61,34,58,35,39);(1,4)(2,10)(3,17)(5,23)(6,24)(7,29)(8,25)(9,30)(11,34)(12,35)(13,22)(14,36)(15,39)(16,44)(18,46)(19,47)(20,51)(21,48)(26,53)(27,56)(28,55)(31,58)(32,52)(33,59)(37,60)(38,54)(40,61)(41,45)(42,62)(43,57)(49,64)(50,63);(1,5,8,7)(2,11,14,13)(3,18,21,20)(4,23,25,29)(6,16,28,27)(9,31,33,32)(10,34,36,22)(12,37,38,26)(15,40,42,41)(17,46,48,51)(19,43,50,49)(24,44,55,56)(30,58,59,52)(35,60,54,53)(39,61,62,45)(47,57,63,64)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26605);