Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C22×C4 |
| direct product | C61×#8113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #95 |
| commutator | C23 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C22×C61 | C2×#95 |
| prime radical | C24×C61 | C22×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C42×C61 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(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);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,22,24,25)(20,26)(21,27)(23,29,30,31)(28,32);(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 1101);