Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#19012 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(2,6)(4,13)(7,15)(8,9)(10,11)(14,16)(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);(1,2,5,9)(3,7,12,16)(4,8,13,6)(10,15,11,14)(20,27)(22,28)(25,31)(29,32);(1,3)(2,7)(4,10)(5,12)(6,14)(8,15)(9,16)(11,13)(17,18)(19,23)(20,28)(21,24)(22,27)(25,32)(26,30)(29,31);(1,4,5,13)(2,8,9,6)(3,10,12,11)(7,15,16,14)(17,19,21,26)(18,23,24,30)(20,25,27,31)(22,29,28,32)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12000);