Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9303 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#9304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #386 |
| 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 | |
436 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)(11,15,20,30)(12,22)(13,25)(14,17)(16,23)(18,27)(19,33,34,40)(21,37,36,29)(24,32)(26,31)(28,38,39,35)(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);(1,2,3,5,4,6,7,8)(9,11)(10,15,17,30)(12,19)(13,21)(14,20)(16,28,31,39)(18,29,25,37)(22,33,32,40)(23,35)(24,34)(26,38)(27,36);(1,3,4,7)(2,5,6,8)(9,12,14,24)(10,16,17,31)(11,19,20,34)(13,26,27,23)(15,28,30,39)(18,32,25,22)(21,38,36,35)(29,40,37,33)3871 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 2292);