Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#7726 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C22×D4×C79 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C43×C79 ◁ G | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,6,5,8,4,7)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,26,28,32,29,31,37,39)(27,33,35,40,34,30,36,38)(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);(2,6)(3,4)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,30)(28,36)(29,34)(31,33)(32,40)(35,37)(38,39);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,31)(27,34)(32,39)(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, 714);