Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#21663 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C22×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | C2×C4×D4 |
| prime radical | #532×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,20,16,21,15,19)(11,22,25,17,27,34,26,35)(12,28,30,38,31,37,13,18)(23,33,32,40,29,36,24,39)(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);(1,3,4,7)(2,5,6,8)(9,11,15,26,16,27,14,25)(10,17,20,34,21,35,19,22)(12,29,13,32,31,23,30,24)(18,36,28,39,38,33,37,40);(2,6)(5,8)(9,12,14,30,16,31,15,13)(10,18,19,37,21,38,20,28)(11,23,25,32,27,29,26,24)(17,33,22,39,35,36,34,40);(9,13,14,12,16,30,15,31)(10,18,20,28,21,38,19,37)(11,24,25,23,27,32,26,29)(17,33,34,40,35,36,22,39)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14651);