Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#7934 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C61 |
| Frattini, cosocle | C2×C42 | C23×C61 |
| socle | C22×C61 | C2×#95 |
| prime radical | C24×C61 | C42 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C42×C61 ◁ G | |
596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,8,5,6,3,7)(9,10)(11,19,21,15)(12,16)(13,18)(14,17)(20,24,22,23)(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);(2,6)(3,4)(9,11,13,22,14,21,12,20)(10,15,18,24,17,19,16,23);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 922);