Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#17543 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C22×C61 | C22×C22⋊C4 |
| prime radical | C61×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C42×C61 ◁ G | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,6,7,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23);(1,3,5,4)(2,6,7,8)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 10531);