Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#13154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #442 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| prime radical | C23×C61 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C22×C4×C61 ◁ C2×#97×C61 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,4)(5,6,10,15,11,17,29,16)(7,18,20,13,22,31,21,32)(8,23,24,35,26,12,25,30)(9,27,28,14,19,33,36,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(5,7,19,8,10,21,36,25,11,22,9,26,29,20,28,24)(6,12,14,31,16,30,27,32,17,23,34,18,15,35,33,13);(2,4)(5,8,9,21,10,25,28,22,11,26,19,20,29,24,36,7)(6,13,14,23,16,31,27,35,17,32,34,12,15,18,33,30)2074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6142);