Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61779 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | C79×#3493 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #525×C79 ◁ G | |
1678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 127 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26)(19,23,29,39)(20,31,34,41)(21,35,36,25)(24,32,42,33)(27,43,45,48)(28,40,30,38)(37,46,47,44)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,29)(18,23,26,39)(20,27,34,45)(21,30,36,28)(24,37,42,47)(25,40,35,38)(31,43,41,48)(32,46,33,44);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,27)(21,32)(22,34)(23,37)(25,41)(26,42)(28,44)(29,45)(30,46)(31,35)(33,36)(38,43)(39,47)(40,48);(3,11)(6,9)(10,16)(13,15)(17,21,22,36)(18,25,26,35)(19,28,29,30)(20,32,34,33)(23,38,39,40)(24,41,42,31)(27,44,45,46)(37,43,47,48);(2,8)(6,9)(7,14)(13,15)(19,29)(23,39)(27,45)(28,30)(37,47)(38,40)(43,48)(44,46)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54767);