Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33769 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×#41×C79 ◁ C2×#131×C79 ◁ G | |
576 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,24)(13,18)(14,16)(15,21)(17,26)(19,23)(22,29)(25,33)(27,37)(28,32)(30,35)(31,40)(34,39)(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);(1,3)(2,5)(4,8)(6,7)(9,11,13,21)(10,15,18,20)(12,25,26,22)(14,23,28,35)(16,30,32,19)(17,33,24,29)(27,39,40,36)(31,34,37,38);(3,8)(5,7)(10,16)(11,21)(14,23)(15,19)(17,31)(18,32)(20,30)(22,25)(24,37)(27,36)(28,35)(29,34)(33,38)(39,40);(2,6)(5,7)(9,12,13,26)(10,17,18,24)(11,22,21,25)(14,27,28,40)(15,29,20,33)(16,31,32,37)(19,34,30,38)(23,36,35,39)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26757);