Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#11999 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #106 |
| commutator | C8 | C22×C8×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| socle | C22×C61 | C2×#106 |
| prime radical | C23×C61 | #106 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 101 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20,23,34)(12,17)(13,26)(14,29)(15,19)(16,22,31,36)(18,27)(21,33,24,38)(25,32)(28,39)(30,35,40,37)(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);(9,11,13,22,14,24,27,37,15,23,28,36,12,21,25,35)(10,16,18,20,17,30,26,33,19,31,32,34,29,40,39,38);(9,12,15,14)(10,17,19,29)(11,21,23,24)(13,25,28,27)(16,30,31,40)(18,26,32,39)(20,33,34,38)(22,35,36,37)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4987);