Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#26047 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C4×C4⋊C4×C61 ◁ G | |
758 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 4 generators on 89 points:
(2,3)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,19)(17,20)(21,28)(22,26)(23,25)(24,27)(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)(13,15,16,22)(14,18,19,26)(17,21,24,23)(20,28,27,25);(5,7,8,12)(6,9,10,11)(14,19)(15,21)(17,24)(18,25)(22,23)(26,28);(1,3,4,2)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 19035);