Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,23,27,22,39,38,49,25,30,37,47,24,29,21,28)(20,31,35,26,41,44,40,43,36,42,32,46,34,48,33,45);(14,15,16,17)(18,20,22,34,25,36,24,41)(19,26,29,43,30,46,39,45)(21,32,23,40,38,35,37,33)(27,42,28,48,47,31,49,44);(18,21,24,37,25,38,22,23)(19,27,39,49,30,47,29,28)(20,32,41,33,36,35,34,40)(26,42,45,44,46,31,43,48)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6261);