Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#8677 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #380 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C23×C79 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 points:
(2,5)(3,7)(6,8)(9,10)(11,19)(12,24,26,16)(13,27)(14,18)(15,22)(17,20)(21,35,37,31)(23,32)(25,30,34,39)(28,36,38,40)(29,33)(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);(9,11,20,33,14,23,13,22)(10,15,27,32,18,29,17,19)(12,21,34,40,26,37,25,36)(16,28,39,35,24,38,30,31);(1,2,4,6)(3,8,7,5)(9,12,14,26)(10,16,18,24)(11,21,23,37)(13,25,20,34)(15,28,29,38)(17,30,27,39)(19,31,32,35)(22,36,33,40)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 1665);