Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C2×D4 |
| commutator | C2×C4 | C22×C8×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C8×D4 |
| prime radical | C2×C13×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C13×C16 ◁ G | |
332 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)(18,19,24,30,23,28,21,39,25,31,40,27,22,29,33,46)(20,32,37,48,36,45,34,42,38,43,49,47,35,26,41,44);(18,20,33,41,22,35,40,49,25,38,21,34,23,36,24,37)(19,26,39,48,29,43,30,44,31,45,46,47,28,32,27,42);(15,17)(18,21,22,24,25,33,23,40)(19,27,28,46,31,30,29,39)(20,34,35,37,38,41,36,49)(26,42,32,47,45,44,43,48)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6112);