Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33460 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C2×#508×C79 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #117×C79 ◁ #405×C79 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2,4,3)(5,6,11,16)(7,14,9,23)(8,13,22,17)(10,25,27,29)(12,24,30,19)(15,34,35,21)(18,32,20,31)(26,33,36,28)(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);(5,7,11,9)(6,12,16,30)(8,18,22,20)(10,26,27,36)(13,28,17,33)(14,29,23,25)(15,32,35,31)(19,34,24,21);(1,3,4,2)(5,8,11,22)(6,13,16,17)(7,18,9,20)(10,21,27,34)(12,28,30,33)(14,32,23,31)(15,29,35,25)(19,36,24,26);(5,9,11,7)(6,14,16,23)(8,18,22,20)(10,24,27,19)(12,29,30,25)(13,31,17,32)(15,33,35,28)(21,36,34,26)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26448);