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×#25803 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
1434 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)(7,17)(8,22)(9,24)(10,16)(11,15)(12,20)(13,25)(14,18)(19,33)(21,30)(23,36)(26,29)(27,35)(28,32)(31,34)(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);(5,7,10,20,11,21,18,8)(6,12,14,17,15,22,16,30)(9,19,25,34,26,35,32,23)(13,27,24,31,28,33,29,36);(1,2,3,4)(5,8,11,20)(6,12,15,22)(7,18,21,10)(9,23,26,34)(13,27,28,33)(14,17,16,30)(19,32,35,25)(24,31,29,36);(5,9,18,32,11,26,10,25)(6,13,16,29,15,28,14,24)(7,19,8,23,21,35,20,34)(12,27,30,36,22,33,17,31)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 18791);