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×#13134 |
| 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 | |
324 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)(15,16)(18,19,24,29,22,28,25,40)(20,30,33,47,32,26,23,38)(21,34,37,46,35,49,41,42)(27,39,44,31,43,48,45,36);(14,15,17,16)(18,20,25,23,22,32,24,33)(19,26,40,47,28,30,29,38)(21,31,41,39,35,36,37,48)(27,42,45,49,43,46,44,34);(14,16,17,15)(18,21,22,35)(19,27,28,43)(20,31,32,36)(23,39,33,48)(24,37,25,41)(26,42,30,46)(29,44,40,45)(34,47,49,38)754 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6122);