Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#25143 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C79×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(3,4)(5,6,9,13,10,14,23,30)(7,11,16,25,17,26,34,36)(8,18,21,29,22,28,20,12)(15,31,33,27,19,35,32,24)(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);(2,4)(5,7,9,16,10,17,23,34)(6,11,13,25,14,26,30,36)(8,19,21,32,22,15,20,33)(12,27,18,35,29,24,28,31);(5,8,10,22)(6,12,14,29)(7,15,17,19)(9,20,23,21)(11,24,26,27)(13,28,30,18)(16,32,34,33)(25,35,36,31);(2,4)(5,9,10,23)(6,13,14,30)(7,16,17,34)(8,20,22,21)(11,25,26,36)(12,28,29,18)(15,32,19,33)(24,35,27,31)3397 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 18131);