Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#17403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×C61×#532 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C42×C61 ◁ G | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,24,21,23)(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);(2,6)(3,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24);(2,6)(7,8)(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 10391);