Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C22×C4 |
| direct product | C61×#13240 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C61 | C2×D8 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 3 generators on 101 points:
(2,4)(5,6,7,8)(9,10)(11,22)(12,27)(13,30)(14,32)(15,33)(16,21)(17,24)(18,28)(19,23)(20,31)(25,35)(26,40)(29,37)(34,38)(36,39)(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);(1,2,3,4)(6,8)(9,11,23,38,15,26,13,24,16,12,28,39,31,29,14,25)(10,17,32,40,20,34,18,22,21,35,30,37,33,36,19,27);(9,12,28,38,15,29,14,24,16,11,23,39,31,26,13,25)(10,17,30,37,20,34,19,27,21,35,32,40,33,36,18,22)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6228);