Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60682 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C22×C61 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×#117×C61 ◁ G | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 101 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,18)(14,28)(15,19)(17,26)(20,33)(21,30)(22,31)(23,37)(25,32)(27,38)(29,36)(34,40)(35,39)(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);(2,5)(6,8)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,31)(19,30)(24,33)(25,35)(26,36)(27,34)(28,37)(32,39)(38,40);(9,12,14,26,15,27,13,25)(10,17,18,24,19,32,28,38)(11,20,23,36,21,34,22,35)(16,29,31,33,30,39,37,40);(1,3,4,7)(2,6,5,8)(11,21)(16,30)(20,34)(22,23)(29,39)(31,37)(33,40)(35,36);(9,13,15,14)(10,18,19,28)(11,22,21,23)(12,25,27,26)(16,31,30,37)(17,24,32,38)(20,35,34,36)(29,33,39,40)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53670);