Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C29 |
| direct product | C61×#60170 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator | C4 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C2×C61×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×#117×C61 ◁ G | |
2742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 81 points:
(1,2,4,3)(5,6)(7,12)(8,13)(9,18)(10,14)(11,15)(16,19)(17,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(5,7)(6,11)(8,15)(9,17)(10,16)(12,18)(13,19)(14,20);(1,3,4,2)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,20)(14,19);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,19,20,12);(6,14)(11,20)(12,19)(13,18)5002 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53158);