Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C29 |
| direct product | C61×#60455 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C61×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×#41×C61 ◁ G | |
2678 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)(5,6,10,14)(7,15,17,29)(8,12,19,26)(9,21,22,13)(11,23,25,18)(16,28,30,36)(20,33,32,27)(24,34,35,31)(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);(5,7,9,18,10,17,22,23)(6,11,13,15,14,25,21,29)(8,16,20,31,19,30,32,34)(12,24,27,28,26,35,33,36);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,28)(17,30)(18,31)(21,33)(22,32)(23,34)(25,35)(29,36);(8,19)(12,26)(16,30)(20,32)(24,35)(27,33)(28,36)(31,34);(7,17)(11,25)(15,29)(16,30)(18,23)(24,35)(28,36)(31,34)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53443);