Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33459 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#508×C79 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #117×C79 ◁ #405×C79 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2,4,3)(5,6)(7,16)(8,11)(9,23)(10,15)(12,17)(13,20)(14,21)(18,29)(19,28)(22,26)(24,32)(25,35)(27,36)(30,34)(31,33)(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);(6,11)(8,20)(9,21)(12,24)(13,14)(15,26)(16,29)(18,34)(19,33)(22,35)(23,25)(27,28)(30,31)(32,36);(1,3,4,2)(7,17)(12,16)(18,32)(19,33)(24,29)(27,30)(28,31)(34,36);(5,7,10,17)(6,12,15,16)(8,18,22,32)(9,19,21,33)(11,24,26,29)(13,27,25,30)(14,28,23,31)(20,34,35,36)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26447);