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×#33671 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C79×#2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
554 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)(11,21)(12,17)(13,18)(14,16)(15,23)(19,26)(20,25)(22,31)(24,34)(27,32)(28,38)(29,36)(30,40)(33,39)(35,37)(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);(2,5)(6,8)(9,11,13,23)(10,15,18,21)(12,22,25,36)(14,24,27,38)(16,28,32,34)(17,29,20,31)(19,30,33,35)(26,37,39,40);(10,16)(12,25)(14,26)(15,28)(17,33)(18,32)(19,20)(21,34)(22,36)(24,37)(27,39)(29,35)(30,31)(38,40);(1,3,4,7)(2,6,5,8)(9,12,13,25)(10,17,18,20)(11,22,23,36)(14,26,27,39)(15,29,21,31)(16,19,32,33)(24,37,38,40)(28,30,34,35)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26659);