Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#33697 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C22×D8 |
| prime radical | C79×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
598 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,17)(13,19)(14,21)(15,20)(16,23)(22,30)(24,36)(25,32)(26,37)(27,33)(28,35)(29,38)(31,39)(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);(1,3,4,7)(2,5,6,8)(9,11,15,25,13,24,29,28,16,26,31,40,30,27,14,12)(10,17,21,33,22,34,39,37,23,35,38,36,19,32,20,18);(2,6)(5,8)(9,12,14,27,30,40,31,26,16,28,29,24,13,25,15,11)(10,18,20,32,19,36,38,35,23,37,39,34,22,33,21,17);(9,13,16,30)(10,19,23,22)(11,24,26,27)(12,25,28,40)(14,15,29,31)(17,32,35,34)(18,36,37,33)(20,38,39,21)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26685);