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×#13286 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C2×#102 |
| prime radical | C23×C13 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20)(17,18)(19,21)(22,23,28,33,29,34,27,45)(24,35,39,47,40,52,38,50)(25,32,26,44,43,46,42,31)(30,36,48,53,49,41,51,37);(15,18)(19,20)(22,24,26,37,28,39,43,36,29,40,42,53,27,38,25,41)(23,30,46,52,45,51,44,47,34,49,32,35,33,48,31,50);(14,16,17,21)(15,19,18,20)(22,25,27,42,29,43,28,26)(23,31,33,32,34,44,45,46)(24,36,38,37,40,41,39,53)(30,47,48,52,49,50,51,35)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6274);