Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18669 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | #532×C79 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
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)(20,27)(23,29)(24,28)(25,31)(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);(1,3,5,13)(2,6,9,15)(4,11,14,12)(7,10,8,16)(20,27)(22,28)(25,31)(29,32);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,15,16)(17,18)(19,23)(20,28)(21,24)(22,27)(25,32)(26,30)(29,31);(1,4,5,14)(2,8,9,7)(3,12,13,11)(6,10,15,16)(17,19,21,26)(18,23,24,30)(20,25,27,31)(22,29,28,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11657);