Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#11111 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #127 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C22×C61 | #392 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C23×C4×C61 ◁ G | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2,3,4)(5,6)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,22)(18,26)(20,30)(23,31)(24,27)(25,29)(28,35)(32,34)(33,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(5,7,9,19,11,20,10,24)(6,12,14,17,16,27,15,30)(8,18,22,33,23,35,25,34)(13,28,21,36,29,26,31,32);(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,21)(15,31)(16,29)(17,32)(19,33)(20,35)(24,34)(27,28)(30,36)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4099);