Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#11492 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×C22⋊C8 |
| prime radical | #532×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,6,4,5,7,8)(9,10,12,17,14,19,27,35,15,20,28,36,13,18,26,34)(11,21,22,38,23,32,39,33,24,37,25,40,29,16,30,31)(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);(2,5)(3,7)(9,11)(10,16)(12,25)(13,29)(14,23)(15,24)(17,33)(18,37)(19,21)(20,32)(22,28)(26,39)(27,30)(31,36)(34,38)(35,40);(9,12,14,27,15,28,13,26)(10,17,19,35,20,36,18,34)(11,22,23,39,24,25,29,30)(16,31,21,38,32,33,37,40)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 4480);