Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C61 · Schur multiplier (H2) = C23 |
| direct product | C61×#13386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C61×#3054 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C22×M16×C61 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 81 points:
(2,3)(5,6)(7,10)(8,17)(9,12)(11,15)(13,16)(14,18)(19,20)(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);(1,2)(3,4)(7,14)(10,18)(11,19)(15,20);(1,3,4,2)(6,10)(7,15)(8,16)(11,13)(12,18)(14,20)(17,19)2074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6374);