Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 115 points:
(5,6,12,31,13,19,29,18)(7,20,24,32,25,36,26,15)(8,14,27,35,22,34,23,33)(9,28,21,16,10,17,11,30)(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)(5,7,21,27,12,24,10,22,13,25,11,23,29,26,9,8)(6,14,28,15,18,33,30,36,19,34,17,32,31,35,16,20);(2,4)(5,8,13,22)(6,15,19,32)(7,10,25,9)(11,24,21,26)(12,23,29,27)(14,17,34,28)(16,33,30,35)(18,20,31,36)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6459);