Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#19439 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C61 |
| prime radical | C61×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G | |
624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,6)(3,8,7,4)(10,14)(11,17)(12,21)(16,23)(19,20)(22,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,4,2,5,7,8,6)(9,10,12,14,13,16,21,23)(11,18,19,24,20,15,17,22);(2,6)(3,7)(9,11,13,20)(10,15,16,18)(12,19,21,17)(14,22,23,24);(10,16)(14,23)(15,18)(22,24)2257 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 12427);