Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C61 · Schur multiplier (H2) = C2 |
| direct product | C61×#10610 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23×C61 |
| prime radical | C23×C61 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,8,4,7,3,6)(9,10,14,18)(11,19,20,23)(12,17,13,16)(15,22,24,21)(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);(9,11,12,21,14,20,13,22)(10,15,16,19,18,24,17,23);(1,2,4,7)(3,8,5,6)(10,16)(11,20)(12,13)(15,19)(17,18)(23,24)2257 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3598);