Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#20516 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C61 | C23 |
| commutator | C22 | C22×C42×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle, prime radical | C23×C61 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C61 ◁ G | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 4 generators on 89 points:
(5,6)(7,11)(8,12)(9,10)(13,14,16,20,17,18,23,25)(15,19,21,26,22,24,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(1,2,4,3)(6,10)(11,12);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,18)(19,24)(20,25)(26,28);(5,8,9,7)(6,12,10,11)(13,15,16,21,17,22,23,27)(14,19,20,26,18,24,25,28)6832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13504);