Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61877 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | C2×#41×C79 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #533×C79 ◁ G | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
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)(17,18,22,25)(19,23,29,39)(20,32,35,42)(21,26,36,27)(24,33,40,34)(28,43,44,47)(30,38,31,41)(37,46,48,45)(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);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,28)(21,31)(22,29)(24,37)(25,39)(26,41)(27,38)(30,36)(32,43)(33,46)(34,45)(35,44)(40,48)(42,47);(17,20)(18,24)(19,28)(21,33)(22,35)(23,37)(25,40)(26,32)(27,42)(29,44)(30,45)(31,46)(34,36)(38,47)(39,48)(41,43);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,29)(24,40)(26,27)(28,44)(30,31)(32,42)(45,46);(2,8)(6,9)(7,16)(14,15)(17,21,22,36)(18,26,25,27)(19,30,29,31)(20,33,35,34)(23,38,39,41)(24,32,40,42)(28,45,44,46)(37,47,48,43)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54865);