Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33469 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×#442 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#509×C79 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(2,8,11,23)(3,13)(5,19)(6,14)(9,12,20,21)(10,22,26,32)(15,28)(16,30)(17,29)(24,27,25,31)(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);(1,2)(3,9)(4,10)(5,20)(6,21)(7,11)(8,13)(12,14)(15,24)(16,25)(17,31)(18,26)(19,23)(22,28)(27,29)(30,32);(1,3,14,19,7,5,6,13)(2,9,12,23,11,20,21,8)(4,16,29,28,18,15,17,30)(10,25,27,22,26,24,31,32);(1,4,7,18)(2,10,11,26)(3,15,5,16)(6,17,14,29)(8,22,23,32)(9,24,20,25)(12,27,21,31)(13,28,19,30)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26459);