Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C29 |
| direct product | C61×#61078 |
| 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 | |
2694 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)(5,6,11,15)(7,16,19,26)(8,13,10,23)(9,14,22,31)(12,24,28,17)(18,33,35,30)(20,29,25,32)(21,36,34,27)(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)(6,12)(8,17)(9,21)(10,24)(11,19)(13,26)(14,30)(15,28)(16,23)(18,20)(22,34)(25,35)(27,32)(29,36)(31,33);(2,4)(5,8,11,10)(6,13,15,23)(7,17,19,24)(9,20,22,25)(12,26,28,16)(14,29,31,32)(18,34,35,21)(27,30,36,33);(5,9,11,22)(6,14,15,31)(7,18,19,35)(8,20,10,25)(12,27,28,36)(13,29,23,32)(16,33,26,30)(17,34,24,21);(6,15)(7,19)(13,23)(14,31)(17,24)(18,35)(21,34)(29,32)3355 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54066);