Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60720 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C61×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C4×#41×C61 ◁ G | |
1846 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)(9,10,14,18)(11,15,20,29)(12,22,25,39)(13,26,27,17)(16,31,33,24)(19,34,23,38)(21,37,36,30)(28,40,32,35)(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)(10,15)(12,23)(13,21)(14,20)(16,32)(17,30)(18,29)(19,25)(22,38)(24,40)(26,37)(27,36)(28,33)(31,35)(34,39);(9,12,13,24,14,25,27,31)(10,16,17,22,18,33,26,39)(11,19,21,35,20,23,36,40)(15,28,30,34,29,32,37,38);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,28)(17,30)(18,29)(22,34)(23,25)(24,35)(26,37)(27,36)(31,40)(32,33)(38,39);(11,20)(15,29)(19,23)(21,36)(28,32)(30,37)(34,38)(35,40)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53708);