Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61477 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #529×C79 ◁ G | |
1772 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)(6,8)(9,10,14,19)(11,15,22,30)(12,23,26,33)(13,17,27,18)(16,24,32,25)(20,34,36,40)(21,29,38,31)(28,35,39,37)(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)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,19,30)(12,20,26,36)(13,21,27,38)(16,28,32,39)(17,29,18,31)(23,34,33,40)(24,35,25,37);(9,12)(10,16)(11,20)(13,24)(14,26)(15,28)(17,23)(18,33)(19,32)(21,35)(22,36)(25,27)(29,34)(30,39)(31,40)(37,38);(2,5)(6,8)(9,13,14,27)(10,17,19,18)(11,21,22,38)(12,24,26,25)(15,29,30,31)(16,23,32,33)(20,35,36,37)(28,34,39,40);(1,3,4,7)(2,6,5,8)(11,22)(15,30)(20,36)(21,38)(28,39)(29,31)(34,40)(35,37)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54465);