Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#23520 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C22×C79 | C2×#121 |
| prime radical | C22×#135×C79 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ G | |
1350 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)(7,16)(8,21)(9,13)(10,15)(11,19)(12,24)(14,25)(17,33)(18,27)(20,31)(22,32)(23,30)(26,34)(28,35)(29,36)(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)(6,11)(8,22)(9,18)(10,20)(12,28)(13,16)(14,31)(15,27)(17,24)(19,25)(21,26)(23,35)(29,33)(30,32)(34,36);(2,4)(5,8,9,23,10,24,25,36)(6,12,14,30,15,21,13,29)(7,17,19,35,20,22,18,34)(11,26,16,32,27,28,31,33);(5,9,10,25)(6,13,15,14)(7,18,20,19)(8,23,24,36)(11,16,27,31)(12,29,21,30)(17,34,22,35)(26,32,28,33)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16508);