Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13170 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C4×D8 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
278 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)(18,19,23,28,24,30,39,29)(20,25,22,38,32,27,40,42)(21,33,36,47,37,41,46,48)(26,34,43,49,45,31,44,35);(15,17)(18,20,23,22,24,32,39,40)(19,25,28,38,30,27,29,42)(21,34,36,49,37,31,46,35)(26,33,43,47,45,41,44,48);(18,21,32,34,23,36,40,49,24,37,20,31,39,46,22,35)(19,26,38,48,29,44,25,41,30,45,42,47,28,43,27,33)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6158);