Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25×C4 |
| direct product | C79×#25806 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×#578×C79 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
1402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(2,4)(5,6,11,16)(7,17,21,31)(8,12,20,27)(9,23,25,29)(10,15,18,14)(13,28,30,24)(19,33,35,26)(22,32,34,36)(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);(5,7,18,20,11,21,10,8)(6,12,15,31,16,27,14,17)(9,19,28,34,25,35,24,22)(13,26,29,36,30,33,23,32);(1,2,3,4)(5,8)(6,12)(7,18)(9,22)(10,21)(11,20)(13,26)(14,17)(15,31)(16,27)(19,28)(23,32)(24,35)(25,34)(29,36)(30,33);(5,9,18,28,11,25,10,24)(6,13,15,29,16,30,14,23)(7,19,20,34,21,35,8,22)(12,26,31,36,27,33,17,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 18794);