Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C61 · Schur multiplier (H2) = C214 |
| direct product | C24×M16×C61 |
| subgroup | G/H | |
|---|---|---|
| center | C24×C4×C61 | C22 |
| commutator | C2 | C25×C4×C61 |
| Frattini, cosocle | C4 | C26×C61 |
| socle | C25×C61 | C2×C4 |
| prime radical | C26×C61 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C4×C61 ◁ G | |
13310 conjugacy classes of subgroups; only the 11166 normal subgroups are shown.
This group has 13310 conjugacy classes of subgroups — too many to draw. Only its 11166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 6 generators on 77 points:
(1,2)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(3,4);(5,6);(7,8);(9,10,11,14,12,13,15,16);(10,13)(14,16)9760 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 56060);