Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#25429 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C61×#1999 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C4×C22⋊C4×C61 ◁ G | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 85 points:
(5,6,8,7)(9,10)(11,16)(12,22)(13,23)(14,18)(15,21)(17,19)(20,24)(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);(1,2)(3,4)(5,7,8,6)(9,11,19,12,14,20,13,21)(10,15,23,24,18,22,17,16);(2,4)(6,7)(11,20)(12,21)(15,22)(16,24);(9,12,19,20,14,21,13,11)(10,16,23,15,18,24,17,22)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 18417);