Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13162 |
| 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 | C23×C13 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C13×C16 ◁ G | |
290 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)(18,19,23,29,24,28,39,45,25,30,38,27,22,40,21,36)(20,26,33,41,34,42,31,46,35,43,47,44,32,48,37,49);(15,17)(18,20,23,33,24,34,39,31,25,35,38,47,22,32,21,37)(19,26,29,41,28,42,45,46,30,43,27,44,40,48,36,49);(18,21,22,38,25,39,24,23)(19,27,28,36,30,29,40,45)(20,31,32,33,35,37,34,47)(26,41,42,46,43,44,48,49)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6150);