Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#19155 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×#121 |
| prime radical | #532×C79 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ G | |
608 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)(3,10)(4,7)(5,9)(6,12)(8,13)(11,15)(14,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,32)(30,31)(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);(1,3,4,12,5,11,13,16)(2,6,8,10,9,14,7,15)(18,22)(23,31)(24,29)(25,32);(3,11)(6,14)(10,15)(12,16)(17,19,20,26)(18,23,22,31)(21,30,28,27)(24,32,29,25);(1,4,5,13)(2,7,9,8)(3,12,11,16)(6,15,14,10)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12143);