Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#18061 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C42 | C24×C61 |
| socle | C22×C61 | C2×C4×D4 |
| prime radical | C2×#135×C61 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C61×#431 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,7,5,3,6,8)(9,10)(11,20)(12,18)(13,17)(14,19)(15,21)(16,22)(23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(9,11)(10,15)(12,21)(13,23)(14,22)(16,18)(17,20)(19,24);(1,3,4,8,5,2,6,7)(9,11)(10,16)(12,23)(13,21)(14,22)(15,18)(17,24)(19,20);(2,3)(7,8)(9,12,14,13)(10,17,19,18)(11,21,22,23)(15,20,24,16)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 11049);