Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #533×C79 ◁ G | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 119 points:
(2,6)(7,8)(9,10)(11,15)(12,24)(13,17)(14,19)(16,26)(18,28)(20,35)(21,30)(22,33)(23,31)(25,32)(27,34)(29,36)(37,39)(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);(1,2,5,6)(3,7,4,8)(9,11,14,23)(10,15,19,31)(12,20,27,37)(13,22,28,21)(16,29,32,40)(17,33,18,30)(24,35,34,39)(25,38,26,36);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,20)(13,25)(14,27)(15,29)(17,24)(18,34)(19,32)(21,36)(22,38)(23,37)(26,28)(30,39)(31,40)(33,35);(1,4,5,3)(2,7,6,8);(9,13,14,28)(10,17,19,18)(11,21,23,22)(12,25,27,26)(15,30,31,33)(16,24,32,34)(20,36,37,38)(29,39,40,35)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54399);