Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#23601 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | C2×#121 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ G | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(5,6,11,15,10,16,9,14)(7,17,25,28,20,27,19,12)(8,13,22,30,24,31,23,32)(18,33,34,26,35,29,21,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,2)(3,4)(5,7,9,19,10,20,11,25)(6,12,15,17,16,28,14,27)(8,21,23,35,24,34,22,18)(13,29,30,36,31,33,32,26);(2,4)(5,8)(6,13)(7,18)(9,22)(10,24)(11,23)(12,26)(14,30)(15,32)(16,31)(17,33)(19,34)(20,35)(21,25)(27,29)(28,36);(5,9,10,11)(6,14,16,15)(7,19,20,25)(8,22,24,23)(12,27,28,17)(13,30,31,32)(18,34,35,21)(26,29,36,33)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16589);