Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#9213 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #121 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| socle | C22×C79 | #386 |
| prime radical | C79×#3106 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,5,8,6,9,4,14)(3,10,13,15,11,16,12,7)(18,22,24,27)(19,25,26,28)(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)(2,7,9,15)(4,12)(5,13)(6,11)(8,10,14,16)(17,18,20,27,21,24,29,22)(19,23,28,31,26,32,25,30);(3,11)(7,15)(10,16)(12,13)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,29)(27,31)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 2201);