Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C27×C4 |
| direct product | C61×#20584 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×C22⋊C4 |
| commutator | C23 | C23×C4×C61 |
| Frattini, cosocle | C24 | C24×C61 |
| socle | C22×C61 | C22×C22⋊C4 |
| prime radical | C23×#41×C61 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C24×C4×C61 ◁ G | |
4006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(2,4);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13572);