Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#19009 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C79×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(2,7)(3,10)(4,5)(9,13)(11,12)(14,15)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29)(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);(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,14)(12,16)(17,19,22,18)(20,25,28,23)(21,24,31,26)(27,30,32,29);(1,3)(2,8)(4,11)(5,10)(6,12)(7,14)(9,16)(13,15)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,27)(29,31);(1,4,6,5)(2,7,9,13)(3,11,12,10)(8,14,16,15)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11997);