Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C22×C4 |
| direct product | C61×#13241 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×D8 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C4×D8 |
| prime radical | C2×C61×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×C8×C61 ◁ C2×C4×C8×C61 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(2,4)(5,6,11,17,12,19,28,18)(7,14,22,34,23,31,36,35)(8,24,20,13,26,32,21,33)(9,27,29,15,10,30,25,16)(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);(1,2,3,4)(5,7,10,21,28,36,29,26,12,23,9,20,11,22,25,8)(6,13,15,14,17,32,30,34,19,33,16,31,18,24,27,35);(5,8,25,22,11,20,9,23,12,26,29,36,28,21,10,7)(6,14,30,33,18,35,15,32,19,31,27,13,17,34,16,24)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6229);