Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C26 |
| direct product | C61×#22468 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | D42 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×C61×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ G | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 97 points:
(1,2)(3,4)(5,6,10,16,12,8,11,15)(7,17,21,18,22,29,20,13)(9,23,26,32,27,14,25,31)(19,30,24,28,36,33,35,34)(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)(9,24)(10,21)(11,20)(12,22)(14,30)(15,29)(16,17)(19,25)(23,33)(26,36)(27,35)(28,31)(32,34);(2,4)(5,8,12,6)(7,18,22,13)(9,23,27,14)(10,16,11,15)(17,20,29,21)(19,34,36,28)(24,33,35,30)(25,32,26,31);(5,9)(6,14)(7,19)(8,23)(10,26)(11,25)(12,27)(13,28)(15,32)(16,31)(17,33)(18,34)(20,35)(21,24)(22,36)(29,30)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 15456);