Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#12556 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | D8 |
| commutator | C8 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C23×C79 | C2×D8 |
| prime radical | C24×C79 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,4)(9,10)(11,22)(12,18)(13,20)(14,19)(15,30)(16,21)(17,24)(23,32)(25,36)(26,40)(27,37)(28,34)(29,35)(31,38)(33,39)(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);(5,6,7,8)(9,11,14,25,15,26,29,39,16,27,28,38,12,23,13,24)(10,17,20,32,18,31,34,37,21,33,35,40,30,36,19,22);(9,12,16,15)(10,18,21,30)(11,23,27,26)(13,28,29,14)(17,31,33,36)(19,20,34,35)(22,32,37,40)(24,38,39,25)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5544);