Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#62091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #529×C79 ◁ G | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 119 points:
(2,5)(3,7)(9,10)(11,19)(12,23)(13,27)(14,18)(15,22)(16,26)(17,28)(20,35)(21,30)(24,32)(25,31)(29,36)(33,38)(34,39)(37,40)(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);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,30)(18,19)(23,40)(24,36)(25,37)(26,38)(27,34)(28,39)(31,35)(32,33);(9,12,14,25)(10,16,18,32)(11,20,22,37)(13,24,28,26)(15,29,19,33)(17,31,27,23)(21,36,39,38)(30,35,34,40);(2,5)(6,8)(9,13,14,28)(10,17,18,27)(11,21,22,39)(12,24,25,26)(15,30,19,34)(16,31,32,23)(20,36,37,38)(29,35,33,40);(1,3,4,7)(2,6,5,8)(10,18)(11,22)(16,32)(17,27)(20,37)(21,39)(23,31)(36,38)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 55079);