Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C22×C4 |
| direct product | C61×#9668 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #121 |
| commutator | C2×C4 | C22×C8×C61 |
| Frattini, cosocle | C2×C42 | C23×C61 |
| socle | C22×C61 | #390 |
| prime radical | C22×D8×C61 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ G | |
510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,7,5,3,6,8)(9,10,14,18,15,17,13,11)(12,20,22,24,23,19,21,16)(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);(1,3,4,8,5,2,6,7)(9,11,14,17,15,18,13,10)(12,19,22,24,23,20,21,16);(9,12,14,22,15,23,13,21)(10,16,11,19,17,24,18,20)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 2656);