Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#13200 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #127 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1955 |
| prime radical | #525×C79 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,5,10,6,9,19,25,7,11,20,26,4,17,18,30)(3,12,15,29,16,23,22,31,13,28,27,32,14,8,21,24)(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);(1,2,5,10,6,9,19,25,7,11,20,26,4,17,18,30)(3,12,27,32,16,23,21,24,13,28,15,29,14,8,22,31);(1,3,4,14,7,13,6,16)(2,8,9,12,11,23,17,28)(5,15,18,21,20,27,19,22)(10,24,25,29,26,31,30,32)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6188);