Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#11086 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #124 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C22×C61 | #397 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C23×C4×C61 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(5,6,10,16,11,14,9,15)(7,12,20,28,18,29,19,27)(8,21,25,31,23,13,24,32)(17,26,34,35,33,36,22,30)(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);(1,2)(3,4)(5,7)(6,12)(8,22,23,34)(9,19)(10,20)(11,18)(13,30,21,35)(14,29)(15,27)(16,28)(17,24,33,25)(26,31,36,32);(2,4)(5,8)(6,13)(7,17)(9,25)(10,24)(11,23)(12,26)(14,21)(15,32)(16,31)(18,33)(19,34)(20,22)(27,35)(28,30)(29,36)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4074);