Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33735 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C22×D8 |
| prime radical | C79×#2215 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,26)(14,19)(16,24)(17,21)(18,22)(20,33)(25,38)(27,29)(28,34)(30,40)(31,39)(32,36)(35,37)(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)(9,11,14,22)(10,15,19,18)(12,20,25,35)(13,21,27,36)(16,28,31,30)(17,29,32,26)(23,33,38,37)(24,34,39,40);(9,12,21,34,11,20,27,39,14,25,36,40,22,35,13,24)(10,16,26,37,18,30,32,38,19,31,29,33,15,28,17,23);(1,3,4,7)(2,6,5,8)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26723);