Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#12600 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C79 | C23 |
| commutator | C22 | C2×C4×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C23×C79 | C22×C8 |
| prime radical | C24×C79 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C79 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,4,3)(5,6)(7,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,26)(20,27)(21,25)(22,30)(23,28)(24,29)(31,37)(32,35)(33,36)(34,38)(39,40)(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);(6,8)(9,11,12,19,13,20,22,31,14,21,23,32,24,33,34,39)(10,15,17,26,18,27,30,37,16,25,28,35,29,36,38,40);(1,3,4,2)(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5588);