Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#24149 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C61×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G | |
1038 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,14)(13,17)(15,22)(18,23)(19,24)(20,21)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(2,5)(6,7)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22);(9,12,14,10)(11,18,20,24)(13,22,15,17)(16,23,21,19);(1,3,4,8)(2,6,5,7)(9,13,14,15)(10,17,12,22)(11,19,20,23)(16,24,21,18)2623 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 17137);