Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#60685 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| socle | C22×C79 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×#117×C79 ◁ G | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 119 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,23)(13,17)(14,19)(16,25)(18,27)(20,35)(21,29)(22,30)(24,38)(26,33)(28,36)(31,37)(32,39)(34,40)(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);(2,5)(6,8)(9,11)(10,15)(12,24)(13,22)(14,21)(16,32)(17,30)(18,31)(19,29)(20,26)(23,38)(25,39)(27,37)(28,34)(33,35)(36,40);(9,12,13,25,14,26,27,40)(10,16,18,23,19,34,17,33)(11,20,22,36,21,24,37,39)(15,28,31,35,29,32,30,38);(1,3,4,7)(2,6,5,8)(11,21)(15,29)(20,24)(22,37)(28,32)(30,31)(35,38)(36,39);(9,13,14,27)(10,17,19,18)(11,22,21,37)(12,25,26,40)(15,30,29,31)(16,33,34,23)(20,36,24,39)(28,38,32,35)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 53673);