Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13145 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #466 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
314 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)(14,15,17,16)(19,24)(20,29)(21,33)(22,34)(25,35)(26,42)(27,40)(28,39)(30,32)(31,46)(36,43)(37,38)(41,45)(44,47);(14,16,17,15)(18,19,25,40,22,27,21,24,23,28,33,38,34,37,35,39)(20,26,41,49,31,42,30,36,32,44,46,48,45,47,29,43);(18,20,21,30,34,45,25,41,23,32,35,29,22,31,33,46)(19,26,24,36,37,47,40,49,28,44,39,43,27,42,38,48)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6133);