Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#8602 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #117 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| socle | C22×C79 | #380 |
| prime radical | C24×C79 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×#97×C79 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,27)(20,33,35,31)(22,32)(24,30,39,38)(28,34,37,40)(29,36)(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)(9,11,13,21,14,22,27,36)(10,15,17,19,18,29,26,32)(12,20,24,34,25,35,39,40)(16,28,30,31,23,37,38,33);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,31)(21,34)(22,35)(26,38)(27,39)(29,37)(32,33)(36,40)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 1590);