Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#11471 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #99 |
| commutator | C23 | C22×C8×C61 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C61 |
| socle | C22×C61 | C2×#99 |
| prime radical | C2×#120×C61 | C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C4×C61 ◁ C2×C4×D4×C61 ◁ G | |
756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,4,7,5,8,3,6)(9,10)(11,15,16,13)(12,14)(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);(10,13)(14,15);(1,3,5,4)(2,6,8,7)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4459);