Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33464 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C22×D8 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×C4×D8 |
| prime radical | C79×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C2×M16×C79 ◁ #405×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:
(1,2,6,12,8,13,7,28)(3,14,17,40,18,41,29,37)(4,10,21,36,19,34,22,51)(5,23,25,38,27,53,26,11)(9,24,31,45,33,54,32,16)(15,39,44,61,42,60,52,59)(20,47,49,58,46,63,50,35)(30,48,56,62,55,64,57,43)(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)(4,15)(5,24)(6,17)(7,29)(8,18)(10,30)(11,37)(12,31)(13,33)(14,23)(16,26)(19,42)(20,48)(21,44)(22,52)(25,45)(27,54)(28,32)(34,55)(35,59)(36,56)(38,40)(39,47)(41,53)(43,50)(46,64)(49,62)(51,57)(58,61)(60,63);(1,4,6,21,8,19,7,22)(2,10,12,36,13,34,28,51)(3,15,17,44,18,42,29,52)(5,20,25,49,27,46,26,50)(9,30,31,56,33,55,32,57)(11,35,23,47,38,58,53,63)(14,39,40,61,41,60,37,59)(16,43,24,48,45,62,54,64);(4,19)(10,34)(15,42)(20,46)(21,22)(30,55)(35,58)(36,51)(39,60)(43,62)(44,52)(47,63)(48,64)(49,50)(56,57)(59,61)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26452);