Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8790 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C8 |
| direct product | C79×#8793 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #382 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C23×C79 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 points:
(2,7)(3,8)(4,5)(9,10)(11,19,22,32)(12,23,25,16)(13,17)(14,18)(15,21,28,36)(20,33)(24,38,39,29)(26,30)(27,34)(31,35)(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);(1,2,4,7,6,8,5,3)(9,11)(10,15,18,28)(12,20)(13,21)(14,22)(16,27,23,37)(17,19,30,32)(24,34)(25,35)(26,36)(29,31,38,33)(39,40);(1,3,5,8,6,7,4,2)(9,12,13,24,14,25,26,39)(10,16,17,29,18,23,30,38)(11,20,21,34,22,35,36,40)(15,27,19,31,28,37,32,33)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 1781);