Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#33701 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C22×D8 |
| prime radical | C79×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
590 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)(3,5)(4,6)(7,8)(9,10)(11,18)(12,28)(13,19)(14,31)(15,33)(16,23)(17,25)(20,29)(21,32)(22,30)(24,35)(26,36)(27,38)(34,39)(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);(1,3,4,7)(2,5,6,8)(9,11,15,26,13,24,14,25,16,27,29,39,30,40,32,12)(10,17,21,35,22,36,20,18,23,28,31,37,19,34,33,38);(2,6)(5,8)(9,12,14,24,30,39,15,11,16,25,32,40,13,26,29,27)(10,18,20,34,19,35,21,28,23,38,33,36,22,37,31,17);(9,13,16,30)(10,19,23,22)(11,24,27,40)(12,26,25,39)(14,29,32,15)(17,34,28,36)(18,35,38,37)(20,21,33,31)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26689);