Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33676 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
548 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)(3,6)(4,5)(7,8)(9,10,14,20)(11,15,23,31)(12,17,24,19)(13,16,26,33)(18,34,32,25)(21,28,35,30)(22,27,37,39)(29,40,38,36)(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);(2,5)(6,8)(9,11,14,23)(10,15,20,31)(12,21,24,35)(13,22,26,37)(16,27,33,39)(17,28,19,30)(18,29,32,38)(25,36,34,40);(10,16)(12,24)(13,25)(15,27)(17,32)(18,19)(20,33)(21,35)(22,36)(26,34)(28,38)(29,30)(31,39)(37,40);(1,3,4,7)(2,6,5,8)(9,12,14,24)(10,17,20,19)(11,21,23,35)(13,25,26,34)(15,28,31,30)(16,18,33,32)(22,36,37,40)(27,29,39,38)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26664);