Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#11721 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #99 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C2×C4⋊C4 | C23×C61 |
| socle | C22×C61 | C2×#99 |
| prime radical | C24×C61 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C4×C61 ◁ C2×#117×C61 ◁ G | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 85 points:
(2,5,6,7)(9,10,13,18,14,17,19,22)(11,15,21,23,20,24,12,16)(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)(3,7)(4,6)(5,8)(9,11)(10,15,17,24)(12,19)(13,21)(14,20)(16,22,23,18);(9,12,14,21)(10,16,17,23)(11,19,20,13)(15,22,24,18)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4709);