Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C22×C4 |
| direct product | C61×#12674 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C61 | C2×D8 |
| commutator | C8 | C2×C42×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,3)(9,10,11,16,19,23,24,21,13,15,20,22,12,17,18,14)(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);(5,6,7,8)(10,14)(11,18)(12,19)(15,21)(16,17)(20,24)(22,23);(1,3,4,2)(10,15)(14,21)(16,22)(17,23)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5662);