Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C210 |
| direct product | C61×#60399 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C61 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C61 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C61 ◁ C22×D4×C61 ◁ G | |
5128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 976 |
A minimal faithful permutation representation — 5 generators on 77 points:
(6,12)(13,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,15)(12,16);(4,11)(7,14)(9,15)(13,16)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53387);