Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#61883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C61×#3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C4×Q8×C61 ◁ G | |
1958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 85 points:
(2,5)(3,7)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24);(1,2,4,6)(3,8,7,5)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24);(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24);(10,18)(15,19)(16,17)(23,24)3721 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54871);