Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C28 |
| direct product | C2×C61×#2960 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C61 | C23 |
| commutator | C22 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C23×C61 | C23×C4 |
| prime radical | C24×C61 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C61 ◁ G | |
1982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 87 points:
(1,2)(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);(3,4)(5,10)(6,8)(7,9)(11,12,14,18,15,17,21,24)(13,16,19,23,20,22,25,26);(3,5,7,6)(4,8,9,10);(3,6,7,5)(4,8,9,10)(11,13,14,19,15,20,21,25)(12,16,18,23,17,22,24,26);(12,17)(16,22)(18,24)(23,26)6832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53175);