Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C27×C4 |
| direct product | C79×#60809 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | C79×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×#117×C79 ◁ G | |
2034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 5 generators on 119 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18)(11,15,20,29)(12,22,25,31)(13,26,27,17)(16,23,32,24)(19,33,34,40)(21,38,37,30)(28,35,39,36)(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);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,30)(18,29)(22,33)(23,35)(24,36)(25,34)(26,38)(27,37)(31,40)(32,39);(3,8)(5,7)(9,12)(10,16)(11,19)(13,23)(14,25)(15,28)(17,31)(18,32)(20,34)(21,35)(22,26)(24,27)(29,39)(30,40)(33,38)(36,37);(2,6)(5,7)(11,20)(15,29)(19,34)(21,37)(28,39)(30,38)(33,40)(35,36);(9,13,14,27)(10,17,18,26)(11,21,20,37)(12,23,25,24)(15,30,29,38)(16,31,32,22)(19,35,34,36)(28,40,39,33)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 53797);