Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C24×C79 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,4,3)(5,6)(7,8)(9,10,14,20,12,18,29,37,16,22,13,19,15,21,28,36)(11,23,26,33,32,39,30,17,27,34,25,40,24,38,31,35)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,3,4,2)(9,11)(10,17)(12,24)(13,30)(14,31)(15,32)(16,27)(18,33)(19,38)(20,39)(21,40)(22,35)(23,37)(25,29)(26,28)(34,36);(6,8)(9,12,16,15)(10,18,22,21)(11,24,27,32)(13,28,14,29)(17,33,35,40)(19,36,20,37)(23,38,34,39)(25,30,26,31)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5982);