Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#11210 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | #120 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| prime radical | C24×C61 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C23×C4×C61 ◁ G | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 77 points:
(2,3)(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(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);(1,2,4,3)(11,16)(13,14);(1,3,4,2)(10,13)(11,16)(14,15)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4198);