Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C26 |
| direct product | C61×#22467 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | D42 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×C61×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ G | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2)(3,4)(5,6,10,14,11,7,9,15)(8,17,18,12,20,13,19,16)(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);(5,7,11,6)(8,16,20,12)(9,14,10,15)(13,18,17,19);(2,4)(5,8)(6,12)(7,16)(9,18)(10,19)(11,20)(13,15)(14,17);(5,8)(6,13)(7,17)(9,19)(10,18)(11,20)(12,15)(14,16)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 15455);