Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#26652 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×AGL(1,ℤ/8)×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
784 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)(9,10,15,30,14,20,13,19)(11,21,29,16,25,36,24,32)(12,17,27,18,28,35,26,34)(22,37,23,31,39,33,38,40)(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);(9,11)(10,16)(12,23)(13,29)(14,25)(15,24)(17,33)(18,31)(19,36)(20,32)(21,30)(22,27)(26,39)(28,38)(34,40)(35,37);(2,5)(6,8)(9,12)(10,17)(11,22)(13,26)(14,28)(15,27)(16,31)(18,30)(19,34)(20,35)(21,37)(23,29)(24,38)(25,39)(32,40)(33,36);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,23)(13,27)(14,28)(15,26)(16,31)(17,30)(19,35)(20,34)(21,33)(22,29)(24,39)(25,38)(32,40)(36,37)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 19640);