Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#12749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #447 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C2×C61×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×#131×C61 ◁ G | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,4)(5,6,11,16,12,8,10,15)(7,17,21,33,22,34,20,31)(9,24,27,23,28,32,26,14)(13,25,30,36,18,35,29,19)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(5,7)(6,13)(8,18)(9,25)(10,20)(11,21)(12,22)(14,31)(15,29)(16,30)(17,24)(19,26)(23,33)(27,36)(28,35)(32,34);(2,4)(5,8,12,6)(7,18,22,13)(9,23,28,14)(10,15,11,16)(17,19,34,36)(20,29,21,30)(24,26,32,27)(25,33,35,31)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5737);