Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33538 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
1088 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)(4,21)(5,11)(6,26)(7,13)(8,14)(9,18)(10,22)(12,25)(16,43)(17,38)(19,50)(20,42)(23,54)(24,34)(27,40)(28,39)(29,37)(30,36)(31,58)(32,48)(33,49)(35,56)(41,44)(45,51)(46,63)(47,62)(52,53)(55,60)(57,61)(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)(2,9)(4,16)(5,17)(6,27)(7,19)(8,20)(10,31)(11,32)(12,38)(13,40)(14,33)(15,26)(18,25)(21,51)(22,44)(23,55)(24,47)(28,49)(29,48)(30,46)(34,59)(35,62)(36,60)(37,42)(39,50)(41,52)(43,54)(45,53)(56,58)(57,64)(61,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,30)(8,24)(9,31)(11,34)(12,35)(13,21)(14,36)(15,41)(17,44)(18,45)(19,46)(20,47)(25,53)(26,52)(27,55)(28,57)(29,56)(32,59)(33,60)(37,61)(38,62)(39,54)(40,51)(42,63)(43,50)(48,58)(49,64);(1,5,8,7)(2,11,14,13)(3,17,20,19)(4,22,24,30)(6,25,29,28)(9,32,33,40)(10,34,36,21)(12,37,39,26)(15,38,42,50)(16,44,47,46)(18,48,49,27)(23,53,56,57)(31,59,60,51)(35,61,54,52)(41,62,63,43)(45,58,64,55)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26526);