Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#18013 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C22×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×C4×D4 |
| prime radical | C2×#41×C79 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C4×M16×C79 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,21,24)(19,25,28,31)(20,29,30,23)(22,26,32,27)(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);(17,19,21,28)(18,22,24,32)(20,26,30,27)(23,31,29,25);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16);(3,9)(6,12)(10,15)(13,16)(17,20,21,30)(18,23,24,29)(19,26,28,27)(22,31,32,25)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11001);