Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33470 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×#442 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
632 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)(3,14,18,44)(4,19)(5,11)(6,25)(7,30)(8,13)(9,17,34,53)(10,20)(12,28)(15,46,49,64)(16,39,31,45)(21,40)(22,36)(23,38)(24,41)(26,33,51,35)(27,37)(29,42)(32,57,63,60)(43,47,62,50)(48,61,52,54)(55,56)(58,59)(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)(5,16)(6,26)(7,31)(8,18)(10,32)(11,33)(12,39)(13,34)(14,42)(17,24)(19,54)(20,47)(21,57)(22,49)(23,50)(25,44)(27,60)(28,53)(29,51)(30,35)(36,61)(37,46)(38,63)(40,64)(41,45)(43,55)(48,58)(52,56)(59,62);(1,4,7,20,8,22,5,23)(2,10,11,19,13,38,30,36)(3,15,31,47,18,49,16,50)(6,27,28,58,29,21,24,56)(9,32,33,54,34,63,35,61)(12,40,25,59,41,37,42,55)(14,43,39,64,44,62,45,46)(17,52,26,60,53,48,51,57);(1,5,8,7)(2,11,13,30)(3,16,18,31)(4,20,22,23)(6,24,29,28)(9,33,34,35)(10,36,38,19)(12,25,41,42)(14,39,44,45)(15,47,49,50)(17,51,53,26)(21,56,27,58)(32,61,63,54)(37,59,40,55)(43,46,62,64)(48,57,52,60)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26458);