Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C79×#3283 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C4×D4×C79 ◁ C8×D4×C79 ◁ G | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,23)(13,27)(14,19)(16,24)(17,21)(18,22)(20,33)(25,40)(26,38)(28,35)(29,31)(30,36)(32,39)(34,37)(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)(2,5)(4,8)(6,7)(9,11,14,22)(10,15,19,18)(12,20,26,25)(13,21,29,37)(16,30,32,35)(17,31,34,27)(23,33,38,40)(24,36,39,28);(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,33)(18,35)(19,32)(21,24)(22,25)(23,27)(29,36)(31,38)(34,40)(37,39);(3,8)(5,7)(9,11,14,22)(10,15,19,18)(12,20,26,25)(13,21,29,37)(16,30,32,35)(17,31,34,27)(23,33,38,40)(24,36,39,28)4345 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26857);