Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33590 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#508×C79 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #117×C79 ◁ #405×C79 ◁ G | |
694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 115 points:
(2,4)(5,6,10,15)(7,12,16,26)(8,11,21,25)(9,22,20,14)(13,29,24,19)(17,23,30,34)(18,33,31,28)(27,36,35,32)(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)(7,16)(12,26)(17,30)(18,31)(23,34)(27,35)(28,33)(32,36);(6,11)(8,19)(9,20)(12,23)(13,14)(15,25)(17,32)(18,31)(21,29)(22,24)(26,34)(27,28)(30,36)(33,35);(5,7,10,16)(6,12,15,26)(8,17,21,30)(9,18,20,31)(11,23,25,34)(13,27,24,35)(14,28,22,33)(19,32,29,36)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26578);