Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C4 |
| direct product | C79×#9893 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C79 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C43×C79 ◁ G | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,4,6)(3,7,8,5)(10,13,14,15)(17,18,20,24,21,23,28,29)(19,22,26,31,25,30,27,32)(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);(9,10,11,15,12,14,16,13)(17,19,20,27,21,25,28,26)(18,22,29,31,23,30,24,32);(1,3,4,8)(2,5,6,7)(19,25)(22,30)(26,27)(31,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 2881);