Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#13011 |
| 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 | |
476 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)(5,6,11,18,12,19,9,16,13,20,29,35,21,30,10,17)(7,15,23,14,24,32,8,26,25,31,28,33,27,36,22,34)(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);(1,2,3,4)(5,7,21,24,13,25,12,27)(6,14,30,26,20,33,19,34)(8,10,28,29,22,9,23,11)(15,16,32,18,31,17,36,35);(5,8)(6,15)(7,10)(9,25)(11,27)(12,23)(13,22)(14,16)(17,33)(18,26)(19,36)(20,31)(21,28)(24,29)(30,32)(34,35)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5999);