Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61501 |
| 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 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 119 points:
(2,5)(6,8)(9,10,14,18)(11,20,23,15)(12,25,28,36)(13,17,30,19)(16,27,34,29)(21,37,26,40)(22,35,24,32)(31,38,33,39)(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)(3,6)(4,5)(7,8)(9,11)(10,15)(12,26)(13,24)(14,23)(16,33)(17,35)(18,20)(19,32)(21,28)(22,30)(25,37)(27,38)(29,39)(31,34)(36,40);(9,12)(10,16)(11,21)(13,27)(14,28)(15,31)(17,25)(18,34)(19,36)(20,33)(22,38)(23,26)(24,39)(29,30)(32,37)(35,40);(2,5)(6,8)(9,13,14,30)(10,17,18,19)(11,22,23,24)(12,27,28,29)(15,32,20,35)(16,25,34,36)(21,38,26,39)(31,37,33,40);(1,3,4,7)(2,6,5,8)(11,23)(15,20)(21,26)(22,24)(31,33)(32,35)(37,40)(38,39)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54489);