Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C29 |
| direct product | C2×C61×#2949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator | C4 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C23×D4×C61 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×#117×C61 ◁ G | |
2870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 83 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(3,4,6,5)(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21);(7,9,11,17)(8,12,13,21)(10,19,20,16)(14,22,18,15);(3,5,6,4)(7,8)(9,12)(10,14)(11,13)(15,16)(17,21)(18,20)(19,22);(8,13)(12,21)(14,18)(15,22)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53151);