Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#8694 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #380 |
| commutator | C42 | C22×C4×C61 |
| Frattini, cosocle | C2×C42 | C23×C61 |
| prime radical | #121×C61 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,7,5,3,8,6)(9,10,14,17,15,12,13,23)(11,18,21,19,22,24,20,16)(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);(9,11,13,20,15,22,14,21)(10,16,17,18,12,19,23,24);(1,3)(2,5)(4,6)(7,8)(9,12)(10,15)(11,19)(13,23)(14,17)(16,22)(18,21)(20,24)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 1682);