Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61931 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | C79×#3220 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #533×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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,23)(20,32)(21,25)(22,27)(24,34)(26,36)(28,43)(29,38)(30,39)(31,41)(33,40)(35,42)(37,45)(44,48)(46,47)(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);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19)(18,23)(20,28)(21,31)(22,29)(24,37)(25,41)(26,39)(27,38)(30,36)(32,43)(33,46)(34,45)(35,44)(40,47)(42,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,22,35)(18,24,27,40)(19,28,29,44)(21,33,36,34)(23,37,38,47)(25,32,26,42)(30,45,31,46)(39,48,41,43);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(19,29)(23,38)(28,44)(30,31)(37,47)(39,41)(43,48)(45,46);(2,10)(7,11)(8,9)(15,16)(17,21,22,36)(18,25,27,26)(19,30,29,31)(20,33,35,34)(23,39,38,41)(24,32,40,42)(28,45,44,46)(37,48,47,43)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54919);