Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#60682 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| socle | C22×C79 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×#117×C79 ◁ G | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 119 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,17)(14,19)(16,24)(18,26)(20,33)(21,28)(22,29)(25,31)(27,35)(30,36)(32,37)(34,39)(38,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,5)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,27)(17,29)(18,30)(19,28)(23,33)(24,35)(25,34)(26,36)(31,39)(32,38)(37,40);(9,12,13,24,14,25,26,37)(10,16,18,23,19,32,17,31)(11,20,22,35,21,34,36,40)(15,27,30,33,28,38,29,39);(1,3,4,7)(2,6,5,8)(11,21)(15,28)(20,34)(22,36)(27,38)(29,30)(33,39)(35,40);(9,13,14,26)(10,17,19,18)(11,22,21,36)(12,24,25,37)(15,29,28,30)(16,31,32,23)(20,35,34,40)(27,39,38,33)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 53670);