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×#33649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C24×C79 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,23,38)(12,24,26,16)(13,27,29,40)(15,30,33,21)(17,34,36,25)(20,35,22,37)(28,31,39,32)(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);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,20)(13,28)(14,23)(16,31)(17,35)(18,33)(19,27)(21,25)(22,26)(24,32)(29,39)(30,34)(36,37)(38,40);(9,12,14,26)(10,16,18,24)(11,20,23,22)(13,25,29,34)(15,31,33,32)(17,27,36,40)(19,37,38,35)(21,39,30,28)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26637);