Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33626 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C22×D8 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
546 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)(15,22)(17,25)(18,26)(21,33)(23,39)(24,36)(28,35)(29,32)(30,37)(31,38)(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);(1,3,4,8)(2,5,6,7);(9,11)(10,15)(12,21)(13,28)(14,24)(16,30)(17,33)(18,35)(19,31)(20,27)(22,25)(23,26)(29,37)(32,36)(34,39)(38,40);(9,12,14,26)(10,16,19,18)(11,21,24,23)(13,25,29,40)(15,30,31,35)(17,32,34,27)(20,33,36,39)(22,37,38,28)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26614);