Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13125 |
| 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×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C13×C16 ◁ 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)(18,19,24,30,23,28,21,39,25,31,41,27,22,29,33,46)(20,32,37,47,36,44,40,45,38,43,48,49,35,26,34,42);(18,20,33,34,22,35,41,48,25,38,21,40,23,36,24,37)(19,26,39,47,29,43,30,42,31,44,46,49,28,32,27,45);(15,17)(18,21,22,24,25,33,23,41)(19,27,28,46,31,30,29,39)(20,34,35,48,38,40,36,37)(26,42,32,47,44,45,43,49)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6113);