Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#22468 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×#578×C79 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
1316 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)(5,6,10,15,12,16,11,8)(7,17,21,30,22,18,20,13)(9,24,27,23,28,14,26,32)(19,31,25,29,35,33,34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(5,7)(6,13)(8,18)(9,25)(10,21)(11,20)(12,22)(14,31)(15,17)(16,30)(19,26)(23,36)(24,33)(27,35)(28,34)(29,32);(5,8,12,15)(6,10,16,11)(7,18,22,17)(9,23,28,32)(13,21,30,20)(14,27,24,26)(19,31,35,33)(25,36,34,29);(5,9)(6,14)(7,19)(8,23)(10,27)(11,26)(12,28)(13,29)(15,32)(16,24)(17,33)(18,31)(20,34)(21,25)(22,35)(30,36)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15456);