Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| prime radical | C79×#3506 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
930 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)(3,10,13,15)(4,7,14,8)(6,11,16,12)(17,18,22,26)(19,27,28,31)(20,25,21,24)(23,30,32,29)(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);(1,3,5,13)(2,6,9,16)(4,11,14,12)(7,10,8,15)(17,19,22,28)(18,23,26,32)(20,30,21,29)(24,31,25,27);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,16,15)(18,24)(20,21)(23,31)(25,26)(27,32)(29,30);(1,4,5,14)(2,7,9,8)(3,12,13,11)(6,15,16,10)(19,28)(23,32)(27,31)(29,30)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11605);