Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#13196 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C2×D4 |
| commutator | C2×C4 | C22×C8×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C8×D4 |
| prime radical | C2×C13×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C13×C16 ◁ G | |
430 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)(18,19,24,30,25,29,40,46,22,28,32,27,23,41,21,38)(20,31,36,45,37,47,33,48,34,26,42,43,35,44,39,49);(14,15,16,17)(18,20,32,42,25,37,21,39,22,34,24,36,23,35,40,33)(19,26,27,45,29,44,38,48,28,31,30,43,41,47,46,49);(18,21,23,32,22,40,25,24)(19,27,29,38,28,30,41,46)(20,33,35,36,34,39,37,42)(26,43,44,49,31,45,47,48)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6184);