Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33876 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C4×D4×C79 ◁ C8×D4×C79 ◁ G | |
564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,20)(12,16)(13,27)(14,19)(15,23)(17,25)(18,26)(21,36)(22,33)(24,40)(28,35)(29,32)(30,37)(31,38)(34,39)(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);(1,3)(2,5)(4,8)(6,7)(11,21)(15,30)(20,36)(22,24)(23,37)(28,38)(31,35)(33,40);(9,11,14,21)(10,15,19,30)(12,22,26,24)(13,28,29,38)(16,31,18,35)(17,33,34,40)(20,32,36,27)(23,39,37,25);(3,8)(5,7)(9,12,14,26)(10,16,19,18)(11,22,21,24)(13,25,29,39)(15,31,30,35)(17,32,34,27)(20,33,36,40)(23,38,37,28)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26864);