Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#22535 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C61 |
| Frattini, cosocle | C24 | C24×C61 |
| prime radical | C2×C61×#553 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4⋊C4×C61 ◁ G | |
1172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,6)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(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);(1,2,5,6)(3,7,4,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(1,4,5,3)(2,7,6,8)3355 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 15523);