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×#13152 |
| 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)(14,15)(16,17)(18,19,24,30,25,32,43,31)(20,33,37,46,21,38,41,49)(22,42,44,28,34,29,23,45)(26,40,27,39,48,36,47,35);(18,20,34,39,43,41,44,40,25,21,22,35,24,37,23,36)(19,26,28,33,30,27,29,46,32,48,45,38,31,47,42,49);(15,17)(18,21,22,39,43,37,23,40,25,20,34,35,24,41,44,36)(19,27,28,46,30,48,29,38,32,47,45,49,31,26,42,33)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6140);