Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C27×C4 |
| direct product | C61×#60815 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×#117×C61 ◁ G | |
2108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 85 points:
(1,2)(3,7)(4,6)(5,8)(10,15)(11,19)(12,14)(16,23)(20,24)(21,22)(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);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,20,21,16)(12,17,14,18)(19,23,22,24);(3,8)(5,7)(9,11)(10,16)(12,19)(13,21)(14,22)(15,20)(17,24)(18,23);(2,6)(5,7)(9,12,13,14)(10,17,15,18)(11,19,21,22)(16,24,20,23);(10,15)(16,20)(17,18)(23,24)2989 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53803);