Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33468 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #442 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×#442 |
| prime radical | C79×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×#97×C79 ◁ G | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,7,13)(3,14)(4,19,6,22)(5,11,8,24)(9,26)(10,20,12,21)(15,29)(16,31)(17,30)(18,32)(23,28)(25,27)(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)(2,9)(4,15)(5,16)(6,23)(7,18)(8,25)(10,27)(11,28)(12,31)(13,30)(14,19)(17,21)(20,26)(22,32)(24,29);(1,4,5,20,7,6,8,21)(2,10,24,22,13,12,11,19)(3,15,16,26,18,23,25,17)(9,27,29,32,30,31,28,14);(1,5,7,8)(2,11,13,24)(3,16,18,25)(4,20,6,21)(9,28,30,29)(10,19,12,22)(14,31,32,27)(15,26,23,17)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26456);