Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #532×C79 ◁ G | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,23)(13,27)(14,19)(16,24)(17,29)(18,22)(20,33)(21,34)(25,38)(26,31)(28,30)(32,35)(36,39)(37,40)(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)(12,24,26,20)(13,28,29,21)(16,31,33,23)(17,34,27,30)(25,35,39,37)(32,36,40,38);(9,12,14,26)(10,16,19,33)(11,20,22,24)(13,25,29,39)(15,23,18,31)(17,32,27,40)(21,35,28,37)(30,36,34,38);(3,8)(5,7)(13,29)(17,27)(21,28)(25,39)(30,34)(32,40)(35,37)(36,38);(9,13,14,29)(10,17,19,27)(11,21,22,28)(12,25,26,39)(15,30,18,34)(16,32,33,40)(20,35,24,37)(23,36,31,38)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54212);