Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33613 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C79×#3061 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×#41×C79 ◁ C2×#131×C79 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(3,4)(5,6,10,13)(7,16,19,8)(9,23,25,35)(11,26,24,15)(12,27,28,18)(14,29,30,17)(20,22,34,32)(21,33,31,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(5,7,17,33,11,20,25,27,10,19,29,36,24,34,9,18)(6,12,23,32,15,21,30,8,13,28,35,22,26,31,14,16);(2,4)(6,13)(8,16)(12,28)(14,30)(15,26)(21,31)(22,32)(23,35);(5,8)(6,7)(9,21)(10,16)(11,22)(12,29)(13,19)(14,18)(15,20)(17,28)(23,33)(24,32)(25,31)(26,34)(27,30)(35,36)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26601);