Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33604 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
690 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)(8,12)(9,22)(10,15)(11,20)(13,17)(14,21)(18,29)(19,30)(23,27)(24,35)(25,32)(26,36)(28,33)(31,34)(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);(5,7,17,32,8,19,23,35,10,18,33,36,21,34,9,20)(6,11,22,31,14,26,28,29,15,24,27,30,12,25,13,16);(2,4)(7,18)(11,24)(16,29)(19,34)(20,35)(25,26)(30,31)(32,36);(5,8,10,21)(6,12,15,14)(7,19,18,34)(9,17,23,33)(11,25,24,26)(13,27,28,22)(16,30,29,31)(20,32,35,36)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26592);