Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61350 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | C79×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #533×C79 ◁ G | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 119 points:
(2,5)(6,7)(9,10,14,18)(11,19,21,15)(12,24,26,33)(13,28,29,17)(16,25,34,27)(20,36,37,40)(22,31,23,32)(30,38,35,39)(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);(1,2,4,5)(3,7,8,6)(9,11,14,21)(10,15,18,19)(12,20,26,37)(13,23,29,22)(16,30,34,35)(17,32,28,31)(24,40,33,36)(25,39,27,38);(9,12)(10,16)(11,20)(13,25)(14,26)(15,30)(17,33)(18,34)(19,35)(21,37)(22,38)(23,39)(24,28)(27,29)(31,40)(32,36);(1,3,4,8)(2,6,5,7)(11,21)(15,19)(20,37)(22,23)(30,35)(31,32)(36,40)(38,39);(9,13,14,29)(10,17,18,28)(11,22,21,23)(12,25,26,27)(15,31,19,32)(16,33,34,24)(20,38,37,39)(30,40,35,36)4345 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54338);