Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C4×#41×C61 ◁ G | |
1870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 101 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,24)(13,17)(14,19)(15,20)(16,30)(18,28)(22,36)(23,33)(25,38)(26,34)(27,39)(29,32)(31,40)(35,37)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(2,5)(6,8)(9,11,15,13)(10,17,20,21)(12,22,27,25)(14,29,16,23)(18,31,34,35)(19,33,30,32)(24,38,39,36)(26,40,28,37);(9,12,16,28,15,27,14,26)(10,18,19,24,20,34,30,39)(11,22,23,37,13,25,29,40)(17,31,33,38,21,35,32,36);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,17,20,21)(12,25,27,22)(14,23,16,29)(18,31,34,35)(19,33,30,32)(24,38,39,36)(26,37,28,40);(9,14,15,16)(10,19,20,30)(11,23,13,29)(12,26,27,28)(17,32,21,33)(18,24,34,39)(22,37,25,40)(31,36,35,38)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53705);