Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #447 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C2×#472×C79 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,4,3)(5,6,8,13,23,33,35,27,10,15,24,26,9,14,21,11)(7,16,17,25,18,31,34,32,20,30,22,36,19,12,28,29)(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);(1,3,4,2)(6,11)(8,21)(9,23)(12,25)(13,14)(15,27)(16,29)(17,28)(18,19)(22,34)(24,35)(26,33)(30,32)(31,36);(5,7)(6,12)(8,22)(9,19)(10,20)(11,25)(13,32)(14,30)(15,31)(16,33)(17,24)(18,23)(21,34)(26,29)(27,36)(28,35)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5984);