Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#61221 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C61×#3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C61×#532 ◁ G | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 101 points:
(1,2,4,6)(3,7,8,5)(9,10,14,17)(11,15,20,31)(12,22,25,34)(13,26,28,18)(16,27,33,24)(19,35,23,40)(21,38,37,30)(29,39,32,36)(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);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,23)(13,21)(14,20)(16,32)(17,31)(18,30)(19,25)(22,40)(24,39)(26,38)(27,36)(28,37)(29,33)(34,35);(9,12,14,25)(10,16,17,33)(11,19,20,23)(13,27,28,24)(15,29,31,32)(18,34,26,22)(21,39,37,36)(30,40,38,35);(3,8)(5,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,29)(17,31)(18,30)(22,35)(23,25)(24,36)(26,38)(27,39)(28,37)(32,33)(34,40);(10,17)(11,20)(16,33)(18,26)(19,23)(21,37)(22,34)(36,39)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54209);