Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C27×C4 |
| direct product | C61×#60802 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C42×C61 ◁ G | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,18)(13,15)(14,17)(16,21)(20,24)(22,23)(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);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(17,18)(21,22);(3,8)(5,7)(9,11)(10,16)(12,21)(13,20)(14,22)(15,23)(17,19)(18,24);(2,6)(5,7)(9,12,13,14)(10,17,15,18)(11,21,20,22)(16,19,23,24);(10,15)(16,23)(17,18)(19,24)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53790);