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×#13151 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C4×D8 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
332 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)(14,15)(16,17)(18,19,23,30,25,31,24,40)(20,32,35,43,37,48,36,46)(21,29,38,42,39,47,22,28)(26,41,44,49,45,34,27,33);(18,20,21,33,23,35,38,41,25,37,39,49,24,36,22,34)(19,26,42,48,40,27,29,43,31,45,28,32,30,44,47,46);(15,17)(18,20,21,33,23,35,38,41,25,37,39,49,24,36,22,34)(19,27,28,46,40,45,47,48,31,44,42,43,30,26,29,32)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6139);