Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#12648 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #442 |
| commutator | C8 | C2×C42×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
422 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)(18,19,22,27)(20,30)(21,36,23,39)(24,41,37,29)(25,33)(26,40,28,38)(31,46)(32,45)(34,49)(35,43)(42,47)(44,48);(15,17)(18,20,23,33,24,34,40,47,22,35,21,32,37,31,38,48)(19,25,28,43,29,44,39,49,27,45,26,30,41,42,36,46);(20,31,35,34)(21,23)(25,42,45,44)(26,28)(30,46,43,49)(32,48,33,47)(36,39)(38,40)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5636);