Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#23649 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C2×#124 |
| 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 | |
892 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)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,21,18)(12,27,15,26)(16,22)(17,23)(19,25)(20,29,31,28)(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);(1,3)(2,7)(4,15)(5,18)(6,14)(8,20)(9,24)(10,21)(11,25)(12,16)(13,19)(17,28)(22,31)(23,27)(26,32)(29,30);(1,4,5,17,6,16,19,30)(2,8,9,23,10,22,25,32)(3,12,13,28,14,15,18,29)(7,20,11,26,21,31,24,27);(4,16)(8,22)(12,15)(17,30)(20,31)(23,32)(26,27)(28,29)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16637);