Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#21601 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C61 | C23 |
| commutator | C23 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C23×C61 | C23×C4 |
| prime radical | C24×C61 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C61 ◁ G | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,18,21,17,20,26,16)(15,22,25,24,23,19,27,28)(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);(5,7,9,8)(6,10,11,12)(13,15,17,23)(14,19,20,22)(16,24,21,28)(18,25,26,27);(2,4)(15,23)(19,22)(24,28)(25,27);(5,8,9,7)(6,10,11,12)(13,16,26,20,17,21,18,14)(15,24,27,22,23,28,25,19)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 14589);