Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C29 |
| direct product | C61×#60183 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C61 | C22×D4 |
| commutator | C4 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C2×C61 | C22×C4×D4 |
| prime radical | C61×#3584 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C61 ◁ C2×#131×C61 ◁ G | |
2612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 5 generators on 93 points:
(1,2,6,11,7,3,12,10)(4,8,16,22,17,13,25,21)(5,9,19,24,18,14,26,23)(15,20,29,31,28,27,32,30)(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)(2,7)(4,13)(5,14)(6,10)(8,17)(9,18)(11,12)(15,27)(16,21)(19,23)(20,28)(22,25)(24,26)(29,30)(31,32);(1,4,6,16,7,17,12,25)(2,8,11,22,3,13,10,21)(5,15,19,29,18,28,26,32)(9,20,24,31,14,27,23,30);(1,5)(2,9)(3,14)(4,15)(6,19)(7,18)(8,20)(10,23)(11,24)(12,26)(13,27)(16,29)(17,28)(21,30)(22,31)(25,32);(5,18)(9,14)(15,28)(19,26)(20,27)(23,24)(29,32)(30,31)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53171);