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×#13240 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 3328 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,20,21)(22,23)(24,35)(25,40)(26,43)(27,45)(28,46)(29,34)(30,37)(31,41)(32,36)(33,44)(38,48)(39,53)(42,50)(47,51)(49,52);(14,15,16,17)(19,21)(22,24,36,51,28,39,26,37,29,25,41,52,44,42,27,38)(23,30,45,53,33,47,31,35,34,48,43,50,46,49,32,40);(22,25,41,51,28,42,27,37,29,24,36,52,44,39,26,38)(23,30,43,50,33,47,32,40,34,48,45,53,46,49,31,35)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6228);