Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#17504 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C42 | C24×C61 |
| socle | C22×C61 | C22×C22⋊C4 |
| prime radical | C61×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C42×C61 ◁ G | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 77 points:
(2,6)(3,4)(7,8)(9,10,13,15,14,11,12,16)(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)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,15,11,16);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,11,15)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 10492);