Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#12843 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #466 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C2×C13×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C13×#131 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1664 |
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)(18,19,23,27,24,21,35,28)(20,29,33,45,34,47,49,46)(22,37,38,26,39,41,30,36)(25,40,42,44,31,48,43,32);(18,20,30,48,23,33,22,32,24,34,38,40,35,49,39,44)(19,25,41,45,28,43,26,29,21,31,37,46,27,42,36,47);(15,17)(18,21)(19,24)(20,31)(22,36)(23,27)(25,34)(26,39)(28,35)(29,44)(30,37)(32,47)(33,42)(38,41)(40,45)(43,49)(46,48)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5831);