Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#13242 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C4×D8 |
| prime radical | C2×#508×C79 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,3,4)(5,6,11,18,13,19,12,32)(7,20,23,36,25,15,24,35)(8,26,21,14,28,33,22,34)(9,29,27,17,10,31,30,16)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(2,4)(5,7,9,21,11,23,27,28,13,25,10,22,12,24,30,8)(6,14,17,15,32,26,29,36,19,34,16,20,18,33,31,35);(5,8,27,23,12,22,9,7,13,28,30,24,11,21,10,25)(6,15,31,14,18,35,16,33,19,20,29,34,32,36,17,26)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6230);