Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#9361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #386 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C23×C4 | C23×C79 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 103 points:
(1,2)(3,5,7,8)(4,6)(9,10,13,17,14,18,12,15)(11,19,21,16,22,23,20,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3)(2,5,6,8)(4,7)(10,15,18,17)(16,19,24,23);(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,23)(18,24)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 2349);