Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#21548 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C2×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | C2×C4×D4 |
| prime radical | C79×#3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C23×C8×C79 ◁ G | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,16,20,25,17,29,19,12)(8,13,22,27,23,28,21,34)(18,30,33,35,31,36,32,26)(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);(5,7,9,19,11,17,10,20)(6,12,14,16,15,25,24,29)(8,18,21,32,23,31,22,33)(13,26,27,30,28,35,34,36);(2,4)(7,17)(12,25)(16,29)(18,31)(19,20)(26,35)(30,36)(32,33);(5,8,10,22,11,23,9,21)(6,13,14,27,15,28,24,34)(7,18,20,33,17,31,19,32)(12,26,16,30,25,35,29,36)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14536);