Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#23646 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | C2×#121 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,20,13,18)(11,16,23,32,24,33,22,31)(12,25,27,36,29,35,28,17)(21,34,26,39,37,40,38,30)(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)(10,16)(12,26)(13,22)(14,23)(15,24)(17,34)(18,31)(19,32)(20,33)(21,28)(25,39)(27,37)(29,38)(30,35)(36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,30)(18,25)(19,35)(20,36)(22,26)(23,38)(24,37)(31,34)(32,40)(33,39);(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,29,28)(16,31,33,32)(17,25,36,35)(21,26,37,38)(30,34,39,40)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16634);