Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#23652 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | C2×#397 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C23×C4×C79 ◁ G | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,10,7,12,6,11)(3,13,16,8,17,24,15,25)(4,18,21,27,22,26,20,9)(14,29,30,32,19,23,31,28)(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)(2,8)(4,19)(5,15)(6,16)(7,17)(9,23)(10,13)(11,24)(12,25)(14,22)(18,32)(20,31)(21,30)(26,28)(27,29);(1,4,6,21,7,22,5,20)(2,9,11,18,12,27,10,26)(3,14,15,30,17,19,16,31)(8,23,13,28,25,29,24,32);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,20,22,21)(8,13,25,24)(9,26,27,18)(14,30,19,31)(23,28,29,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16640);