Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#13262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #459 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
462 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)(15,17)(18,19,22,27,23,38,36,45,25,29,35,44,24,28,21,34)(20,30,31,41,39,46,49,43,33,47,48,42,32,40,37,26);(14,15,16,17)(18,20,25,33)(19,26,29,43)(21,31,36,48)(22,37,35,49)(23,32,24,39)(27,40,44,46)(28,41,38,42)(30,45,47,34);(18,21,24,35,25,36,23,22)(19,27,38,45,29,44,28,34)(20,31,39,49,33,48,32,37)(26,40,42,47,43,46,41,30)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6250);