Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#12988 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | #386 |
| commutator | C2×C4 | C22×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C2×C61 | #1869 |
| prime radical | C61×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×C8×C61 ◁ G | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,6,12,7,10,24,27,8,11,25,29,5,22,14,31)(3,13,17,23,18,28,32,4,16,20,30,19,15,9,26,21)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,14,26,5,15,25,30,8,16,24,32,7,18,6,17)(2,9,27,23,22,20,12,21,11,28,31,19,10,13,29,4);(1,4)(2,3)(5,23)(6,13)(7,19)(8,21)(9,14)(10,18)(11,16)(12,30)(15,22)(17,29)(20,25)(24,28)(26,27)(31,32)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5976);