Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33630 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
542 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)(9,10,13,15)(11,19,21,37)(12,24,25,17)(14,28,27,40)(16,32,30,22)(18,35,31,26)(20,36,23,38)(29,33,39,34)(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)(10,15)(16,30)(17,24)(18,31)(19,37)(28,40)(33,34)(36,38);(9,11)(10,16)(12,20)(13,21)(14,29)(15,30)(17,33)(18,36)(19,28)(22,26)(23,25)(24,34)(27,39)(31,38)(32,35)(37,40);(9,12,13,25)(10,17,15,24)(11,20,21,23)(14,26,27,35)(16,33,30,34)(18,28,31,40)(19,38,37,36)(22,39,32,29)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26618);