Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33438 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C24 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C23×C8 |
| prime radical | #532×C79 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ G | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(3,4)(5,6,8,12,9,13,19,25,10,14,20,26,21,27,32,35)(7,11,17,23,18,24,31,34,15,16,22,29,28,30,33,36)(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);(2,4)(7,15)(11,16)(17,22)(18,28)(23,29)(24,30)(31,33)(34,36);(5,6,8,12,9,13,19,25,10,14,20,26,21,27,32,35)(7,16,17,29,18,30,31,36,15,11,22,23,28,24,33,34);(5,7)(6,11)(8,17)(9,18)(10,15)(12,23)(13,24)(14,16)(19,31)(20,22)(21,28)(25,34)(26,29)(27,30)(32,33)(35,36)7900 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26426);