Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#32883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C23×#41×C61 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C24×C4×C61 ◁ G | |
3926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,17)(11,18,20,23)(12,16,13,21)(15,22,24,19)(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)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,4)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18);(6,8)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 25871);