Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13129 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | #124 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1924 |
| prime radical | C2×C13×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,24,20,25,36,27,18,23,26,30,33,39,41,43)(16,17,22,31,28,32,38,42,29,21,34,35,40,37,44,45);(14,16,26,22,20,28,41,38,18,29,19,34,33,40,36,44)(15,21,24,31,25,37,27,42,23,17,30,35,39,32,43,45);(14,17,18,21)(15,22)(16,27)(19,35,26,31)(20,32,33,37)(23,34)(24,40)(25,38)(28,30)(29,43)(36,45,41,42)(39,44)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6117);