Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C27 |
| direct product | C61×#20919 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C22×D4 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C22×C61 | C2×C4×D4 |
| prime radical | C22×#135×C61 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C22×M16×C61 ◁ G | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 97 points:
(2,4)(5,6)(7,17,19,27)(8,22,25,30)(9,15)(10,16)(11,14)(12,20,28,21)(13,24,29,23)(18,32)(26,33)(31,34)(35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2,3,4)(5,7,9,21,11,19,10,20)(6,12,15,27,14,28,16,17)(8,18,24,34,25,33,23,35)(13,26,22,36,29,32,30,31);(5,8)(6,13)(7,18)(9,24)(10,23)(11,25)(12,26)(14,29)(15,22)(16,30)(17,31)(19,33)(20,35)(21,34)(27,36)(28,32);(6,14)(7,19)(13,29)(15,16)(18,33)(20,21)(22,30)(34,35)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13907);