Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C61×#2140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C2×D8 |
| commutator | C8 | C23×C4×C61 |
| Frattini, cosocle | C2×C8 | C24×C61 |
| prime radical | C23×D8×C61 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C23×C4×C61 ◁ C23×C8×C61 ◁ G | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 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);(3,4,5,6)(7,8)(9,14)(10,13)(11,20)(12,16)(15,18)(17,21)(19,22);(8,13)(9,11)(10,18)(14,15)(16,22)(17,19)(20,21);(7,9,12,11)(8,14,16,20)(10,17,19,18)(13,21,22,15)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26365);