Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60195 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C25 |
| commutator | C22 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C22×C61 | C24×C4 |
| prime radical | C2×#41×C61 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ G | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,11,14,16,21,18)(12,15,20,17,19,22,24,23)(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);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,13,16,14,18,21,10)(12,17,20,22,19,23,24,15);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24);(12,19)(15,22)(17,23)(20,24)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53183);