Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#10817 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #117 |
| commutator | C2×C4 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C22×C61 | #382 |
| prime radical | C24×C61 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×#117×C61 ◁ G | |
642 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)(7,17,20,12)(8,22,24,30)(9,15)(10,16)(11,14)(13,23,29,25)(18,32)(19,28,21,27)(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);(5,7,10,21,11,20,9,19)(6,12,16,28,14,17,15,27)(8,18,25,34,24,35,23,33)(13,26,30,32,29,31,22,36);(2,4)(5,8)(6,13,14,29)(7,18)(9,23)(10,25)(11,24)(12,26,17,31)(15,22,16,30)(19,33)(20,35)(21,34)(27,36,28,32)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3805);