Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C27 |
| direct product | C79×#36693 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C79 |
| socle | C23×C79 | C25 |
| prime radical | #392×C79 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 111 points:
(1,2,5,7)(3,6,4,8)(11,15)(14,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,21,26)(18,22,23,25)(20,27,28,32)(24,29,30,31);(2,7)(6,8)(10,13)(11,15)(19,26)(22,25)(27,32)(29,31);(1,4,5,3)(2,8,7,6)(9,11)(10,14)(12,15)(13,16)(18,23)(22,25)(24,30)(29,31);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 29681);