Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#33966 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C61 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C4×Q8×C61 ◁ C8×Q8×C61 ◁ G | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 85 points:
(2,6)(7,8)(9,10,14,18)(11,15,13,22)(12,16,21,24)(17,19,23,20)(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);(1,2,5,6)(3,7,4,8)(9,11,14,13)(10,15,18,22)(12,19,21,20)(16,23,24,17);(1,3,5,4)(2,7,6,8)(10,16)(11,13)(12,20)(15,17)(18,24)(19,21)(22,23);(1,4,5,3)(2,7,6,8)2623 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 26954);