Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#27061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C24×C79 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C4×C22⋊C4×C79 ◁ G | |
672 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,26,28,22)(20,25,30,24)(27,31,29,32)(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);(2,5)(3,8)(17,19,21,28)(18,22,23,26)(20,27,30,29)(24,31,25,32);(1,3,4,8)(2,6,5,7)(10,13)(14,16)(18,23)(19,27)(20,30)(22,31)(26,32)(28,29);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(17,20,21,30)(18,24,23,25)(19,27,28,29)(22,31,26,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 20049);