Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#7906 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C61 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C23×C61 | C2×C22⋊C4 |
| prime radical | C24×C61 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C61 ◁ C22×C42×C61 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,7,9,8,10,11,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26)(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);(1,2,3,4)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 894);