Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61376 |
| 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 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 119 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,20)(12,25)(13,17)(14,18)(15,22)(16,28)(19,29)(21,36)(23,33)(24,31)(26,32)(27,34)(30,39)(35,38)(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);(1,3,5,8)(2,6,4,7)(9,11,14,22)(10,15,18,20)(12,21,27,37)(13,24,29,23)(16,30,32,35)(17,33,19,31)(25,40,34,36)(26,39,28,38);(9,12)(10,16)(11,21)(13,26)(14,27)(15,30)(17,25)(18,32)(19,34)(20,35)(22,37)(23,38)(24,39)(28,29)(31,36)(33,40);(1,4,5,2)(3,6,8,7)(11,22)(15,20)(21,37)(23,24)(30,35)(31,33)(36,40)(38,39);(9,13,14,29)(10,17,18,19)(11,23,22,24)(12,26,27,28)(15,31,20,33)(16,25,32,34)(21,38,37,39)(30,36,35,40)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54364);