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) = C22 |
| direct product | C79×#8692 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #380 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | #121×C79 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,7,6,8,5,4)(9,10,14,18,15,19,13,28)(11,20,23,31,24,35,22,16)(12,25,26,32,27,17,29,33)(21,36,37,40,38,34,39,30)(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);(1,3,6,5)(2,7,8,4)(9,11,14,23,15,24,13,22)(10,16,28,35,19,31,18,20)(12,21,29,39,27,38,26,37)(17,30,33,36,25,40,32,34);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,32)(19,25)(20,34)(22,37)(23,39)(24,38)(28,33)(31,40)(35,36)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 1680);