Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C61 · Schur multiplier (H2) = C214 |
| direct product | C24×Q16×C61 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C61 | D4 |
| commutator, Frattini, cosocle | C4 | C26×C61 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C61 ◁ C24×C4×C61 ◁ G | |
11166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 6 generators on 85 points:
(1,2)(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);(3,4);(5,6);(7,8);(9,10,13,16)(11,17,19,24)(12,20,22,15)(14,21,23,18);(9,11,13,19)(10,14,16,23)(12,21,22,18)(15,24,20,17)6832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 56067);