Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18313 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C79×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
910 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,9)(3,10,13,16)(4,7,14,8)(6,11,15,12)(18,22)(20,29)(23,31)(24,25)(26,27)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3)(2,6)(4,11)(5,13)(7,10)(8,16)(9,15)(12,14)(17,18)(19,23)(20,24)(21,25)(22,29)(26,31)(27,30)(28,32);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,15,16)(17,19,21,28)(18,23,25,32)(20,26,29,27)(22,30,24,31);(17,20,21,29)(18,24,25,22)(19,26,28,27)(23,31,32,30)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11301);