Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C29 |
| direct product | C61×#60167 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator | C4 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C22×C61 | C23×D4 |
| prime radical | C22×#135×C61 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×#117×C61 ◁ G | |
2706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 97 points:
(1,2,4,3)(5,6)(7,16)(8,13)(9,15)(10,23)(11,14)(12,18)(17,30)(19,27)(20,29)(21,28)(22,33)(24,26)(25,32)(31,35)(34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(5,7)(6,12)(8,17)(9,18)(10,24)(11,19)(13,25)(14,26)(15,27)(16,23)(20,31)(21,32)(22,34)(28,35)(29,36)(30,33);(1,3,4,2)(5,8)(6,13)(7,17)(9,21)(10,22)(11,20)(12,25)(14,29)(15,28)(16,30)(18,32)(19,31)(23,33)(24,34)(26,36)(27,35);(6,14)(8,20)(12,26)(15,23)(16,27)(17,31)(21,22)(32,34);(5,9,11,10)(6,15,14,23)(7,18,19,24)(8,21,20,22)(12,27,26,16)(13,28,29,33)(17,32,31,34)(25,35,36,30)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53155);