Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#23622 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | C2×#396 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C23×C4×C79 ◁ G | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,10,7,11,6,20)(3,12,14,23,16,28,15,8)(4,9,21,25,19,26,18,24)(13,27,30,22,17,31,29,32)(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);(1,3,6,15,7,16,5,14)(2,8,10,12,11,23,20,28)(4,17,18,30,19,13,21,29)(9,22,25,31,26,32,24,27);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,26)(12,27)(14,29)(15,30)(16,17)(20,25)(23,32)(28,31);(1,5,7,6)(2,10,11,20)(3,14,16,15)(4,18,19,21)(8,12,23,28)(9,24,26,25)(13,29,17,30)(22,27,32,31)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16610);