Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#33702 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C79×#3155 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #117×C79 ◁ #405×C79 ◁ G | |
602 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)(3,5)(4,6)(7,8)(10,16)(11,21)(12,13)(14,25)(18,29)(19,30)(20,34)(22,23)(24,36)(26,27)(28,32)(33,39)(35,37)(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);(1,3,4,7)(2,5,6,8)(9,10,13,19,14,20,26,29,15,18,27,34,25,30,12,16)(11,17,21,32,23,33,35,38,24,31,36,40,37,39,22,28);(2,6)(5,8)(9,11,12,22,25,37,27,36,15,24,26,35,14,23,13,21)(10,17,16,28,30,39,34,40,18,31,29,38,20,33,19,32);(10,18)(16,29)(17,31)(19,34)(20,30)(28,38)(32,40)(33,39)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26690);