Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#23609 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C61 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C23×C61 | C22×D4 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C61 ◁ C22×C42×C61 ◁ G | |
1138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,3,4)(9,10)(11,20)(12,21)(13,17)(14,18)(15,19)(16,23)(22,24)(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);(5,6)(7,8)(9,11,13,21,15,16,14,22)(10,12,18,20,19,24,17,23);(2,4)(6,8)(9,12)(10,16)(11,19)(13,23)(14,20)(15,24)(17,22)(18,21);(9,13,15,14)(10,17,19,18)(11,21,16,22)(12,23,24,20)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 16597);