Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18924 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C23×C79 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(2,5)(3,7)(6,8)(9,10,14,18)(11,15,22,32)(12,23,25,33)(13,27,28,17)(16,24,34,26)(19,35,37,39)(20,31,21,30)(29,38,40,36)(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);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,19)(13,20)(14,22)(16,29)(17,30)(18,32)(21,28)(23,39)(24,36)(25,37)(26,38)(27,31)(33,35)(34,40);(9,12,14,25)(10,16,18,34)(11,19,22,37)(13,24,28,26)(15,29,32,40)(17,33,27,23)(20,36,21,38)(30,35,31,39);(1,3,4,7)(2,5,6,8)(9,13,14,28)(10,17,18,27)(11,20,22,21)(12,24,25,26)(15,30,32,31)(16,33,34,23)(19,36,37,38)(29,35,40,39)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11912);