Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C61 · Schur multiplier (H2) = C42 |
| direct product | C4×C61×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C61 | C42 |
| commutator | C4 | C43×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C2×#97 |
| prime radical | C23×C61 | C2×M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C42×C61 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(5,6,7,11,8,13,15,19,9,12,14,20,16,10,17,18);(6,10,12,13)(7,14)(11,19,20,18)(15,17)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5528);