Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#13019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #447 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C22×D8×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
496 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)(5,6,10,16,26,35,25,15,12,18,8,14,11,17,9,27)(7,19,22,33,29,31,21,13,24,32,20,36,23,30,28,34)(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)(5,7)(6,13)(8,21)(9,22)(10,28)(11,29)(12,24)(14,30)(15,19)(16,31)(17,36)(18,34)(20,25)(23,26)(27,32)(33,35);(6,14)(8,25)(9,10)(11,26)(13,30)(15,35)(16,18)(17,27)(19,33)(20,21)(22,28)(23,29)(31,34)(32,36)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6007);