Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#8454 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #379 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| prime radical | C61×#3492 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 85 points:
(2,5,6,7)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(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)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(3,8)(5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 1442);