Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C29 |
| direct product | C61×#61076 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C61×#525 ◁ G | |
2672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 97 points:
(1,2)(3,4)(5,6,10,13)(7,15,17,31)(8,12,21,29)(9,22,25,14)(11,23,27,18)(16,33,34,28)(19,36,35,26)(20,32,24,30)(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)(6,11)(8,19)(9,23)(10,17)(12,28)(13,27)(14,31)(15,22)(16,20)(18,25)(21,35)(24,34)(26,30)(29,33)(32,36);(2,4);(5,8,10,21)(6,12,13,29)(7,16,17,34)(9,24,25,20)(11,26,27,36)(14,32,22,30)(15,33,31,28)(18,35,23,19);(6,13)(7,17)(12,29)(14,22)(16,34)(18,23)(19,35)(30,32)3355 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54064);